If the determinant of the matrix is zero, does its inverse exist?

No, if the determinant of a matrix is zero its inverse does not exist. The inverse of a matrix is found by dividing the adjoint of the matrix by its determinant.

Difference between a singular and non-singular matrix?

If the determinant of a matrix is zero, then it is said to be a singular matrix whereas if the determinant of a matrix is non-zero, then it is said to be nonsingular.

What will be the value of determinant, If the corresponding elements of any two rows (or columns) of a determinant are proportional?

If the corresponding elements of any two rows (or columns) of a determinant are proportional, then the determinant will be zero.

How do you calculate the determinant of a 2x2 matrix?

For a 2x2 matrix [[a, b], [c, d]], the determinant is calculated as ad - bc. This simple formula is the basis for understanding more complex determinant calculations and is often used as a building block in larger determinant expansions.

How does the determinant of a matrix relate to its cofactor matrix?

The determinant of a matrix can be calculated using its cofactors. For any row or column, the determinant is the sum of the products of each element with its cofactor. This relationship, known as the Laplace expansion, provides a recursive method for calculating determinants of larger matrices.

How do determinants relate to the area of a parallelogram?

The absolute value of the determinant of a 2x2 matrix represents the area of the parallelogram formed by the column (or row) vectors of the matrix. This geometric interpretation provides a visual understanding of what determinants represent in two dimensions.

What is the determinant of an identity matrix?

The determinant of an identity matrix is always 1, regardless of its size. This is because an identity matrix has 1s on the main diagonal and 0s elsewhere, and the product of its diagonal elements (which equals its determinant) is always 1. This property is fundamental in understanding how determinants behave.

What is the relationship between the determinant and the trace of a matrix?

While there's no direct formula linking determinant and trace, they are both scalar quantities derived from a matrix and appear in various matrix theorems. For a 2x2 matrix, det(A) = ac - b^2, where a + c is the trace. This relationship becomes more complex for larger matrices but remains an important area of study in linear algebra.

What happens to the determinant if you swap two rows or columns?

When you swap any two rows or columns in a matrix, the sign of the determinant changes. This means the new determinant will be the negative of the original determinant. This property is crucial in understanding how determinants behave under elementary row or column operations.

How does multiplying a row or column by a scalar affect the determinant?

Multiplying any row or column of a matrix by a scalar (constant) multiplies the entire determinant by that scalar. For example, if you double one row of a matrix, its determinant will also double. This property is useful in simplifying determinant calculations.

What is the determinant of a triangular matrix?

The determinant of a triangular matrix (upper or lower) is the product of its diagonal elements. This property makes calculating determinants of triangular matrices much simpler, as you don't need to perform the full determinant expansion.

How does adding a multiple of one row to another row affect the determinant?

Adding a multiple of one row to another row does not change the value of the determinant. This property is extremely useful in simplifying matrices before calculating their determinants, as it allows you to create more zeros in the matrix without altering the final result.

What is the relationship between the determinant of a matrix and its transpose?

The determinant of a matrix is equal to the determinant of its transpose. In other words, det(A) = det(A^T), where A^T is the transpose of matrix A. This property shows that row and column operations have similar effects on the determinant.

What is the significance of a zero determinant?

A zero determinant indicates that the matrix is singular (non-invertible). This means the matrix equations represented by this matrix either have no solution or infinitely many solutions. In geometric terms, it suggests that the transformation represented by the matrix collapses space into a lower dimension.

How does the determinant relate to the inverse of a matrix?

A matrix is invertible if and only if its determinant is non-zero. The inverse of a matrix A is related to its determinant by the formula: A^(-1) = (1/det(A)) * adj(A), where adj(A) is the adjugate of A. This relationship underscores the importance of determinants in matrix algebra.

How does the size of a matrix relate to its determinant?

The size of a matrix directly affects its determinant. Only square matrices (those with an equal number of rows and columns) have determinants. The size of the matrix determines the complexity of calculating its determinant, with larger matrices requiring more steps.

What is a determinant in mathematics?

A determinant is a special number calculated from a square matrix. It provides important information about the matrix, such as whether it's invertible, and is used in solving systems of linear equations and finding the area or volume of geometric shapes.

What is Cramer's Rule and how does it use determinants?

Cramer's Rule is a method for solving systems of linear equations using determinants. It expresses the solution in terms of the ratio of determinants. For a system Ax = b, each variable xi is found by replacing the i-th column of A with b and dividing by det(A). This rule demonstrates a practical application of determinants in solving equations.

What is the connection between determinants and volume in 3D space?

The absolute value of the determinant of a 3x3 matrix represents the volume of the parallelepiped formed by the column (or row) vectors of the matrix. This extends the geometric interpretation of determinants to three dimensions, linking algebraic calculations to spatial concepts.

How does the determinant change when a matrix is multiplied by a scalar?

When a matrix A is multiplied by a scalar k, the determinant is multiplied by k^n, where n is the size of the square matrix. In other words, det(kA) = k^n * det(A). This property shows how scaling affects the determinant and, by extension, the area or volume it represents.

How do determinants relate to eigenvalues?

The determinant of (A - λI), where A is a square matrix, I is the identity matrix, and λ is a scalar, is called the characteristic polynomial of A. The roots of this polynomial are the eigenvalues of A. This connection shows how determinants are crucial in finding eigenvalues, which are important in various applications of linear algebra.

What is the multiplicative property of determinants?

The determinant of a product of matrices is equal to the product of their determinants. In other words, det(AB) = det(A) * det(B) for square matrices A and B. This property simplifies many calculations involving matrix products and is essential in various proofs in linear algebra.

What is the significance of the adjugate matrix in relation to determinants?

The adjugate matrix, also known as the classical adjoint, is closely related to both the inverse and the determinant of a matrix. For a matrix A, its adjugate (adj(A)) multiplied by A gives det(A) times the identity matrix. This relationship is crucial in finding matrix inverses and solving linear systems.

How do determinants help in solving homogeneous systems of equations?

A homogeneous system Ax = 0 has non-trivial solutions if and only if det(A) = 0. This property links the concept of linear independence to determinants, as a zero determinant indicates that the columns (or rows) of A are linearly dependent, allowing for non-zero solutions to the system.

How does the determinant change under matrix similarity transformations?

Similar matrices have the same determinant. If B = P^(-1)AP, where P is invertible, then det(B) = det(A). This property is significant because it shows that certain matrix transformations preserve the determinant, which is useful in studying matrix properties that are invariant under similarity transformations.

What is the connection between determinants and linear transformations?

The determinant of a matrix representing a linear transformation gives the factor by which the transformation scales volumes. A determinant of 1 means the transformation preserves volume, while a negative determinant indicates a change in orientation. This interpretation links algebraic properties of matrices to geometric concepts.

How do determinants relate to the solution of differential equations?

Determinants play a crucial role in solving systems of differential equations, particularly in finding the general solution of homogeneous linear systems. The characteristic equation of the system, whose roots determine the nature of the solution, is found using determinants. This application shows the broader relevance of determinants beyond basic linear algebra.

What is the significance of the determinant in the context of matrix diagonalization?

For a matrix to be diagonalizable, it must have n linearly independent eigenvectors, where n is the size of the matrix. The determinant of the matrix formed by these eigenvectors must be non-zero, ensuring their linear independence. This connection highlights the role of determinants in understanding matrix structure and decomposition.

How does the determinant of a block matrix relate to the determinants of its components?

For a block matrix [[A, B], [C, D]] where A, B, C, D are matrices and A and D are square, if A is invertible, then det([[A, B], [C, D]]) = det(A) * det(D - CA^(-1)B). This formula, known as the Schur complement, is useful in simplifying determinant calculations for larger, structured matrices.

What is the relationship between determinants and matrix rank?

The rank of a matrix is equal to the size of the largest non-zero determinant of any square submatrix. A matrix has full rank if and only if its determinant is non-zero. This connection between rank and determinants is fundamental in understanding the structure and properties of matrices.

How do determinants feature in Cramer's rule for solving systems of linear equations?

Cramer's rule expresses the solution to a system of linear equations Ax = b in terms of determinants. For each variable xi, xi = det(Ai) / det(A), where Ai is the matrix formed by replacing the i-th column of A with the vector b. This rule provides a theoretical method for solving linear systems using determinants, though it's often impractical for large systems.

What is the significance of the determinant in the context of matrix exponentials?

The determinant of the matrix exponential e^A is equal to the exponential of the trace of A: det(e^A) = e^(tr(A)). This property connects determinants to matrix functions and is important in various applications, including differential equations and dynamical systems.

How does the determinant relate to the characteristic polynomial of a matrix?

The characteristic polynomial of a matrix A is defined as det(λI - A), where λ is a variable and I is the identity matrix. The roots of this polynomial are the eigenvalues of A. This relationship demonstrates how determinants are fundamental in finding eigenvalues and understanding the spectral properties of matrices.

What is the role of determinants in defining the cross product in three dimensions?

The cross product of two vectors a and b in three dimensions can be expressed as a determinant:

How do determinants feature in the formula for the inverse of a 2x2 matrix?

For a 2x2 matrix A = [[a, b], [c, d]], its inverse (if it exists) is given by:

What is the connection between determinants and the Cayley-Hamilton theorem?

The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation. The characteristic equation is det(λI - A) = 0, where the left side is the characteristic polynomial. This theorem links determinants to polynomial properties of matrices, demonstrating their role in advanced matrix theory.

How do determinants relate to the concept of linear independence?

A set of vectors is linearly independent if and only if the determinant of the matrix formed by these vectors as columns (or rows) is non-zero. This property provides a concrete test for linear independence using determinants, connecting abstract vector space concepts to computable matrix properties.

What is the significance of the determinant in the context of change of basis?

In a change of basis transformation, the determinant of the transformation matrix represents the scaling factor of volumes in the new basis. A determinant of ±1 indicates that the transformation preserves volumes, which is characteristic of orthogonal transformations. This interpretation links determinants to geometric transformations in linear algebra.

How does the determinant of a matrix product relate to the determinants of its factors?

The determinant of a product of matrices is equal to the product of their determinants: det(AB) = det(A) * det(B). This multiplicative property is crucial in many proofs and calculations in linear algebra, simplifying complex matrix operations.

What is the role of determinants in defining the adjugate matrix?

The adjugate matrix adj(A) is defined using the cofactors of A, where each element is the cofactor of the corresponding element in A, transposed. The relationship A * adj(A) = det(A) * I shows how determinants, adjugates, and matrix inverses are interconnected.

Properties of Determinants

Maths
Determinants
Properties of Determinants

Properties of Determinants

Komal MiglaniUpdated on 02 Jul 2025, 06:36 PM IST

A determinant is a special number that can be determined from a matrix. For a determinant to exist, matrix A must be a square matrix. The determinant of the matrix is denoted by det A or |A|. In real life, we can use determinant in graphic designing, and gaming. Determinants also help us in taking necessary steps in business.

This Story also Contains

What are Determinants?
How to find the Determinant of a Matrix?
Properties of Determinants
Singular and non-singular matrix:

Properties of Determinants

In this article, we will learn the properties of determinants. This category falls under the broader category of matrices, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of thirteen question have been asked on this topic including one in 2018, four in 2019, three in 2020, four in 2021, one in 2022, three in 2023.

What are Determinants?

The determinant of a matrix A is a number which is calculated from the matrix. For a determinant to exist, matrix A must be a square matrix. The determinant of the matrix is denoted by det A or |A|.

How to find the Determinant of a Matrix?

For $2 \times 2$ matrices
$
\mathrm{A}=\left[\begin{array}{ll}
a_1 & a_2 \\
b_1 & b_2
\end{array}\right]
$
then $\operatorname{det} \mathrm{A}$ is :
$
|\mathrm{A}|=\left|\begin{array}{ll}
a_1 & a_2 \\
b_1 & b_2
\end{array}\right|=\mathrm{a}_1 \times \mathrm{b}_2-\mathrm{a}_2 \times \mathrm{b}_1
$

For a $3 \times 3$ matrix determinant can be calculated in the following way :
let $\mathrm{A}=\left[\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right]$
then we find $\operatorname{det} \mathrm{A}$ in following way
$
|A|=a_1\left(b_2 \cdot c_3-b_3 \cdot c_2\right)-a_2\left(b_1 \cdot c_3-c_1 b_3\right)+a_3\left(b_1 c_2-b_2 c_1\right)
$

This same process we follow to evaluate the determinant of the matrix of any order. Notice that we start the first term with the +ve sign then the 2nd with the -ve sign and the 3rd again +ve sign, this sign sequence is followed for any order of matrix.

This whole process is row-dependent, the same process can be done using columns, which means we can select an element along a column delete their row and column compute the determinant of left out matrix, and then multiply it with the element that we select. And we will get the same result as we get while doing the whole process along the row.

Properties of Determinants

Property 1: Interchange Property

The value of the determinant remains unchanged if its rows and columns are interchanged.

For example,

Let, $\Delta=\left|\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right|$
Expanding along the first row, we get
$
\begin{aligned}
& \Delta=\mathrm{a}_1\left|\begin{array}{ll}
b_2 & b_3 \\
c_2 & c_3
\end{array}\right|-\mathrm{a}_2\left|\begin{array}{ll}
b_1 & b_3 \\
c_1 & c_3
\end{array}\right|+\mathrm{a}_3\left|\begin{array}{ll}
b_1 & b_2 \\
c_1 & c_2
\end{array}\right| \\
& \Delta=\mathrm{a}_1\left(\mathrm{~b}_2 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_2\right)-\mathrm{a}_2\left(\mathrm{~b}_1 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_1\right)+\mathrm{a}_3\left(\mathrm{~b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1\right)
\end{aligned}
$

By interchanging the rows and columns of $\Delta$, we get the determinant
$
\Delta^{\prime}=\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right|
$

Expanding $\Delta^{\prime}$ along first column, we get
$
\begin{aligned}
& \Delta^{\prime}=\mathrm{a}_1\left(\mathrm{~b}_2 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_2\right)-\mathrm{a}_2\left(\mathrm{~b}_1 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_1\right)+\mathrm{a}_3\left(\mathrm{~b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1\right) \\
& \Delta=\Delta^{\prime}
\end{aligned}
$

Property 2: Switching Property

If any two rows or two columns of a determinant are interchanged, then the sign of the determinant changes but the numerical value remains unaltered.

For example

Interchanging the first and third rows, the new determinant obtained is given by
$
\Delta^{\prime}=\left|\begin{array}{lll}
c_1 & c_2 & c_3 \\
b_1 & b_2 & b_3 \\
a_1 & a_2 & a_3
\end{array}\right|
$

Expanding along the third row, we get
$
\begin{aligned}
\Delta^{\prime} & =a_1\left(c_2 b_3-c_3 b_2\right)-a_2\left(c_1 b_3-c_3 b_1\right)+a_3\left(b_2 c_1-b_1 c_2\right) \\
& =-\left[a_1\left(b_2 c_3-b_3 c_2\right)-a_2\left(b_1 c_3-b_3 c_1\right)+a_3\left(b_1 c_2-b_2 c_1\right)\right] \\
\Delta & =-\Delta^{\prime}
\end{aligned}
$

Property 3: If there is an interchange of rows or columns twice, then the value of the determinant remains the same.

If $\Delta_n$ is the determinant obtained by $\mathrm{n}$ such successive operations, then
$
\Delta_n=\left\{\begin{array}{cc}
-\Delta, & \text { if } \mathrm{n} \text { is odd } \\
\Delta, & \text { if } \mathrm{n} \text { is even }
\end{array}\right.
$

Property 4: Proportionality (Repetition) Property

If any two rows (or columns) of a determinant are identical (all corresponding elements are the same), then the value of the determinant is zero.

For Example,

If we interchange the identical rows (or columns) of the determinant Δ, then by property 2, Δ changes its sign

Let, $\Delta=\left|\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ a_1 & a_2 & a_3\end{array}\right|=-\left|\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ a_1 & a_2 & a_3\end{array}\right| \quad$ (interchanging row 1 and row 3) $=-\Delta$
[By property 2]
$
\begin{aligned}
2 \Delta & =0 \\
\Delta & =0
\end{aligned}
$

If we interchange the identical rows (or columns) of the determinant Δ, then by property 2, Δ changes its sign

Property 5: Scalar Multiple Property

If each element of a row (or a column) of a determinant is multiplied by a constant k, then the value of the determinant is multiplied by k.

For example

Let, $\Delta=\left|\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right|$
and $\Delta^{\prime}$ be the determinant obtained by multiplying the elements of the first row by $\mathrm{k}$.
$
\Delta^{\prime}=\left|\begin{array}{ccc}
k a_1 & k a_2 & k a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|
$

Expanding along the first row, we get
$
\begin{aligned}
\Delta^{\prime} & =\mathrm{ka}_1\left(\mathrm{~b}_2 c_3-b_3 c_2\right)-\mathrm{ka}_2\left(\mathrm{~b}_1 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_1\right)+\mathrm{ka}_3\left(\mathrm{~b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1\right) \\
& =\mathrm{k}\left[\mathrm{a}_1\left(\mathrm{~b}_2 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_2\right)-\mathrm{a}_2\left(\mathrm{~b}_1 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_1\right)+\mathrm{a}_3\left(\mathrm{~b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1\right)\right] \\
\Delta^{\prime} & =\mathrm{k} \Delta
\end{aligned}
$

Hence,
$
\left|\begin{array}{ccc}
k a_1 & k a_2 & k a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|=\mathrm{k}\left|\begin{array}{lll}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|
$

Note:

By this property, we can take out any common factor from any one row or any one column of a given determinant.
If the corresponding elements of any two rows (or columns) of a determinant are proportional (in the same ratio), then the determinant value is zero.

Property 6: Sum Property

If every element of some row or column of a determinant is expressed as the sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants

For example

$
\left|\begin{array}{ccc}
a_1+x & a_2+y & a_3+z \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|=\left|\begin{array}{ccc}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|+\left|\begin{array}{ccc}
x & y & z \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|
$

Proof:
$
\mathrm{LHS}=\left|\begin{array}{ccc}
a_1+x & a_2+y & a_3+z \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|
$

Expanding along the first row, we get
$
\begin{aligned}
& \Delta=\left(\mathrm{a}_1+\mathrm{x}\right)\left(\mathrm{b}_2 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_2\right)-\left(\mathrm{a}_2+\mathrm{y}\right)\left(\mathrm{b}_1 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_1\right)+\left(\mathrm{a}_3+\mathrm{z}\right)\left(\mathrm{b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1\right) \\
& =\mathrm{a}_1\left(\mathrm{~b}_2 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_2\right)-\mathrm{a}_2\left(\mathrm{~b}_1 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_1\right)+\mathrm{a}_3\left(\mathrm{~b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1\right) \\
& \quad+\mathrm{x}\left(\mathrm{b}_2 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_2\right)-\mathrm{y}\left(\mathrm{b}_1 \mathrm{c}_3-\mathrm{b}_3 \mathrm{c}_1\right)+\mathrm{z}\left(\mathrm{b}_1 \mathrm{c}_2-\mathrm{b}_2 \mathrm{c}_1\right) \\
& =\left|\begin{array}{lll}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|+\left|\begin{array}{ccc}
x & y & z \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|
\end{aligned}
$

Property 7: Property of Invariance

If to each element of any row or column of a determinant, the equimultiples of corresponding elements of other rows (or columns) are added, then the value of the determinant remains the same, i.e., the value of the determinant remains the same if we apply the operation

$
\mathrm{R}_{\mathrm{i}} \rightarrow \mathrm{R}_{\mathrm{i}}+\mathrm{kR}_{\mathrm{j}} \text { or } \mathrm{C}_{\mathrm{i}} \rightarrow \mathrm{C}_{\mathrm{i}}+\mathrm{kC}_{\mathrm{j}}
$

Explanation,
Let, $\Delta=\left|\begin{array}{lll}a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right|$ and $\Delta^{\prime}=\left|\begin{array}{ccc}a_1+k c_1 & a_2+k c_2 & a_3+k c_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{array}\right|$ Here, $\Delta^{\prime}$ is obtained by $R_1 \rightarrow R_1+k R_3$
we can write $\Delta^{\prime}$ as
$
\begin{aligned}
& \text { we can write } \Delta^{\prime} \text { as } \\
& \begin{aligned}
\Delta^{\prime} & =\left|\begin{array}{lll}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|+\left|\begin{array}{ccc}
k c_1 & k c_2 & k c_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right| \\
& =\left|\begin{array}{lll}
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right|+\mathrm{k}\left|\begin{array}{lll}
c_1 & c_2 & c_3 \\
b_1 & b_2 & b_3 \\
c_1 & c_2 & c_3
\end{array}\right| \\
& =\Delta+\mathrm{k} \cdot 0
\end{aligned}
\end{aligned}
$
hence, $\Delta^{\prime}=\Delta$

Property 8: Triangle Property

If each element of a determinant above or below one the principal diagonal of a determinant is zero, then the value of the determinant is the product of the diagonal elements.

I.e.

$\left|\begin{array}{lll}a & f & g \\ 0 & b & h \\ 0 & 0 & c\end{array}\right|=\left|\begin{array}{lll}a & 0 & 0 \\ f & b & 0 \\ g & h & c\end{array}\right|=\left|\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right|=\mathrm{abc}$

Property 9: Factor Property

If a determinant D becomes 0 for x = α, then (x - α) is a factor of Δ.

For example,

If $\Delta=\left|\begin{array}{ccc}x & x^2 & x^3 \\ 4 & 16 & 64 \\ 5 & 9 & 11\end{array}\right|$
When, $\mathrm{x}=4$ the value of $\Delta$ becomes 0 $\because$ at $\mathrm{x}=4, \mathrm{R}_1$ and $\mathrm{R}_2$ are identical. and at $\mathrm{x}=0, \Delta=0$, because all element of $\mathrm{R}_1$ becomes 0 hence, $(x-0)$ and $(x-4)$ are the factors of $\Delta$.

Property 10: All-zero Property

If all the elements of a row or column are zero, then the determinant is zero.

Singular and non-singular matrix:

A square matrix is called a singular matrix if its determinant is 0 otherwise it is called a non-singular matrix. Let's say A is a square matrix then it is singular if |A| = 0, otherwise, it will be non-singular if |A| ≠ 0.

Solved Examples Based on Properties of Determinant

Example 1: Let $\mathrm{P}$ and $\mathrm{p}+2$ be prime numbers and let $
\Delta=\left|\begin{array}{ccc}
\mathrm{p}! & (\mathrm{p}+1)! & (\mathrm{p}+2)! \\
(\mathrm{p}+1)! & (\mathrm{p}+2)! & (\mathrm{p}+3)! \\
(\mathrm{p}+2)! & (\mathrm{p}+3)! & (\mathrm{p}+4)!
\end{array}\right|
$. Then the sum of the maximum values of $\alpha$ and $\beta$ such that $\mathrm{P}^\alpha$ and $(\mathrm{p}+2)^\beta$ divide $\Delta$, is
[JEE MAINS 2022]

Solution:
$
\begin{aligned}
& \Delta=\left|\begin{array}{lll}
\mathrm{P}! & (\mathrm{P}+1)! & (\mathrm{P}+2)! \\
(\mathrm{P}+1)! & (\mathrm{P}+2)! & (\mathrm{P}+3)! \\
(\mathrm{P}+2)! & (\mathrm{P}+3)! & (\mathrm{P}+4)!
\end{array}\right| \\
& \Delta=\mathrm{P})(\mathrm{P}+1)!(\mathrm{P}+2)!\left|\begin{array}{lll}
1 & 1 & 1 \\
\mathrm{P}+1 & \mathrm{P}+2 & \mathrm{P}+3 \\
(\mathrm{P}+2)(\mathrm{P}+1) & (\mathrm{P}+3)(\mathrm{P}+2) & (\mathrm{P}+4)(\mathrm{P}+3)
\end{array}\right| \\
& \Delta=2 \mathrm{P}!(\mathrm{P}+1)!(\mathrm{P}+2)!
\end{aligned}
$
which is divisible by $\mathrm{p}^\alpha \&(\mathrm{p}+2)^\beta$
$
\therefore \alpha=3, \beta=1
$

Hence, the required answer is 4

Example 2: If $\left[\begin{array}{ccc}x-4 & 2 x & 2 x \\ 2 x & x-4 & 2 x \\ 2 x & 2 x & x-4\end{array}\right]=(A+B x)(x-A)^2 \quad$ then the ordered pair $(A, B)$ is equal to :
[JEE MAINS 2018]

Solution:

Property of determinant

If a determinant becomes 0 for $x=a$, then $(x-a)$ is a factor of $D$, in other words, if two rows ( or two columns ) become identical for $x=a$, Then $(x-a)$ is a factor of $D$
we can put values of $x=0$ in both the sides $\left[\begin{array}{ccc}-4 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & -4\end{array}\right]=A\left(-A^2\right)$
$
(-4)^3=A^3 \Rightarrow A=-4
or $x=4$
$
$\begin{aligned} & {\left[\begin{array}{lll}0 & 8 & 8 \\ 8 & 0 & 8 \\ 8 & 8 & 0\end{array}\right]=(A+4 B)(4-A)^2} \\ & -8\left(-8^2\right)+8\left(8^2\right)=(4 B-4)\left(8^2\right)=16 \times 8^2=(4 B-4) 8^2 \\ & B=5\end{aligned}$

(-4,5)

Hence, the required answer is (-4,5)

Example 3: Let $A=\left[a_{i j}\right]$ and $B=\left[b_{i j}\right]$ be two $3 \times 3$ real matrices such that $b_{i j}=(3)^{(i+j-2)} a_{j i}$, where, $\mathrm{i}, \mathrm{j}=1,2,3$. if the determinant of $\mathrm{B}$ is 81 , then the determinant of $\mathrm{A}$ is :
[JEE MAINS 2020]

Solution:
$
\begin{aligned}
& |B|=\left|\begin{array}{lll}
b_{11} & b_{12} & b_{13} \\
b_{21} & b_{22} & b_{23} \\
b_{31} & b_{32} & b_{33}
\end{array}\right| \\
& |B|=\left|\begin{array}{lll}
3^0 a_{11} & 3^1 a_{12} & 3^2 a_{13} \\
3^1 a_{21} & 3^2 a_{22} & 3^3 a_{23} \\
3^2 a_{31} & 3^3 a_{32} & 3^4 a_{33}
\end{array}\right|
\end{aligned}
$

Taking Common $3^2$ from $R_3$ and 3 from $R_2$
$
|B|=3^3\left|\begin{array}{ccc}
3^0 a_{11} & 3^1 a_{12} & 3^2 a_{13} \\
3^0 a_{21} & 3^1 a_{22} & 3^2 a_{23} \\
3^0 a_{31} & 3^1 a_{32} & 3^2 a_{33}
\end{array}\right|
$

Taking Common $3^2$ from $C_3$ and 3 from $C_2$
$
\Rightarrow 81=3^3 \cdot 3 \cdot 3^2|\mathrm{~A}| \Rightarrow 3^4=3^6|\mathrm{~A}| \Rightarrow|\mathrm{A}|=\frac{1}{9}
$

Hence, the required answer is $\frac{1}{9}$

Example 4: Let $
\mathrm{A}=\left(\begin{array}{ccc}
{[x+1]} & {[x+2]} & {[x+3]} \\
{[x]} & {[x+3]} & {[x+3]} \\
{[x]} & {[x+2]} & {[x+4]}
\end{array}\right)
$ where $[t]$ denotes the greatest integer less than or equal to $t$. If $\operatorname{det}(\mathrm{A})=192$, then the set of values of $x$ is the interval :
[JEE MAINS 2023]

Solution:
We know that $[x+I]=[x]+I$ for $I \in$ Integer
$
\begin{aligned}
& \operatorname{det}(A)=\left|\begin{array}{ccc}
{[x]+1} & {[x]+2} & {[x]+3} \\
{[x]} & {[x]+3} & {[x]+3} \\
{[x]} & {[x]+2} & {[x]+4}
\end{array}\right|=192 \\
& R_2 \rightarrow R_2-R_1, R_3 \rightarrow R_3-R_2 \\
& \operatorname{det}(A)=\left|\begin{array}{ccc}
{[x]+1} & {[x]+2} & {[x]+3} \\
-1 & 1 & 0 \\
0 & -1 & 1
\end{array}\right|=192 \\
& \Rightarrow([x]+1)(1-0)-([x]+2)(-1-0)+([x]+3)(1-0)=192 \\
& \Rightarrow 3[x]+6=192 \Rightarrow 3[x]=186 \\
& \Rightarrow[x]=62 \Rightarrow x \in[62,63)
\end{aligned}
$

Hence, the required answer is $[62,63)$

Example 5: If $\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2\end{array}\right|=\frac{9}{8}(103 x+81)$,then $\lambda, \frac{\lambda}{3}$ are the roots of the equation
[JEE MAINS 2023]

Solution:
$
\begin{aligned}
& \left|\begin{array}{ccc}
x+1 & x & x \\
x & x+d & x \\
x & x & x+d^2
\end{array}\right|=\frac{9}{8}(103 x+81) \\
& \text { Put } \mathrm{x}=0 \\
& \left|\begin{array}{ccc}
1 & 0 & 0 \\
0 & \lambda & 0 \\
0 & 0 & \lambda^2
\end{array}\right|=\frac{9}{8} \times 81 \\
& \lambda^3=\frac{9^3}{8} \\
& \lambda=\frac{9}{2} \\
& \frac{\lambda}{3}=\frac{9}{2 \times 3} \Rightarrow \frac{3}{2} \\
& \frac{\lambda}{3}=\frac{3}{2} \\
& 4 x^2-24 x+27=0
\end{aligned}
$

Hence, the required answer is $\frac{3}{2}, \frac{9}{2}$

Frequently Asked Questions (FAQs)

Q: What is the significance of the determinant in the context of Sylvester's criterion for positive definiteness?

Sylvester's criterion states that a symmetric matrix is positive definite if and only if all its leading principal minors (determinants of upper-left submatrices) are positive. This criterion directly uses determinants

Q: How do determinants feature in the study of matrix polynomials?

In the study of matrix polynomials p(A), where A is a square matrix, the determinant of p(A) can often be related to p(λ), where λ are the eigenvalues of A. This relationship, exemplified by det(p(A)) = p(λ1) * p(λ2) * ... * p(λn), connects determinants to more advanced topics in matrix analysis.

Q: What is the role of determinants in defining the characteristic equation of a matrix?

The characteristic equation of a matrix A is det(λI - A) = 0, where λ represents the eigenvalues. This equation directly uses the determinant to find eigenvalues, showcasing how determinants are central to understanding the spectral properties of matrices.

Q: How do determinants relate to the concept of matrix similarity?

Similar matrices A and B (where B = P^(-1)AP for some invertible P) have the same determinant. This property is fundamental in understanding how certain matrix properties remain invariant under similarity transformations, which is crucial in studying matrix canonical forms and classifications.

Q: What is the relationship between determinants and the Gram matrix in linear algebra?

The Gram matrix G of a set of vectors is formed by their inner products. The determinant of G is zero if and only if the vectors are linearly dependent. This connection between determinants and the Gram matrix is crucial in various applications, including least squares problems and orthogonality tests.

Q: How do determinants feature in the formulation of Cramer's rule for complex systems?

Cramer's rule extends to complex systems of equations, where the determinants involved may be complex numbers. The interpretation remains similar: the solution components are ratios of determinants, but now these determinants and the resulting solutions can be complex, adding another layer of interpretation to the role of determinants.

Q: What is the significance of the determinant in the context of Gaussian elimination?

In Gaussian elimination, the determinant of the original matrix is equal to the product of the diagonal elements of the resulting upper triangular matrix (ignoring sign changes from row swaps). This property provides an efficient way to calculate determinants as a byproduct of solving linear systems.

Q: How does the determinant relate to the concept of matrix condition number?

The condition number of a matrix, which measures its sensitivity to numerical operations, is related to the ratio of its largest to smallest singular value. For a 2x2 matrix, this can be expressed in terms of the trace and determinant. A matrix with a determinant close to zero typically has a high condition number, indicating potential numerical instability.

Q: What is the geometric interpretation of the determinant for non-square matrices?

While determinants are defined only for square matrices, the concept can be extended to non-square matrices through the use of exterior algebra. In this context, the determinant-like quantity represents the volume of a parallelotope in higher dimensions, generalizing the geometric interpretation beyond 3D space.

Q: How do determinants feature in the solution of homogeneous linear differential equations?

In solving systems of homogeneous linear differential equations, the characteristic equation is formed by taking the determinant of (λI - A), where A is the coefficient matrix. The roots of this equation determine the form of the general solution, demonstrating the crucial role of determinants in differential equations.

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