Singular Matrix: Definition, Properties, Formula and Examples

Singular Matrix

Edited By Komal Miglani | Updated on Jul 02, 2025 06:34 PM IST

Before we start with the concept of a Singular and non-singular matrix, let’s first understand what is a matrix. A rectangular arrangement of objects (numbers or symbols or any other objects) is called a matrix (plural: matrices). A matrix is only a representation of the symbol, number, or object. It does not have any value. Usually, a matrix is denoted by capital letters. Matrix of order m × n, (read as m by n matrix) means that the matrix has m number of rows and n number of columns. In real life, we use Singular and non-singular matrices to solve systems of linear equations, quantum mechanics, and optics.

This Story also Contains

Determinant of matrix
Singular and non-singular matrix
Properties of Determinants
Solved Examples Based on Singular and Non-Singular Matrices

Singular Matrix

In this article, we will cover the Singular and non-singular matrix. 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 twenty-six questions have been asked on this topic in Jee mains (2013 to 2023), two in 2013, two in 2014, two in 2015, one in 2016, six in 2019, two in 2020, four in 2021, three in 2022, and three in 2023.

Determinant of matrix

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

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 :

$
\text { 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 2nd with the -ve sign and 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.

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.

Properties of Determinants

If A and B are square matrices of same order:

i) The determinant of the transpose of matrix A is equal to the determinant of matrix A. det (A’) = det A

ii) The product of the determinant of matrices AB is equal to the product of the determinant of individual matrices. det (AB) = det (A) det (B)

iii) The determinant of the skew-symmetric matrix of odd order is zero. if A is the skew-symmetric matrix of odd order then |A| = 0.

iv) The determinant of the skew-symmetric matrix of even order is a perfect square. if A is a skew-symmetric matrix of even order then |A| is a perfect square.

v) |kA| = kⁿ |A|, where n is the order of matrix A and k is a constant.

vii) |Aⁿ| = |A|ⁿ, where n belongs to N.

viii) |AB| = |BA|

Solved Examples Based on Singular and Non-Singular Matrices

Example 1: Let $A_1, A_2, A_3$ be the three A.P. with the same common difference d and having their first terms as $\mathrm{A}, \mathrm{A}+1, \mathrm{~A}+2$, respectively. Let $a, b, c$ be the $7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}$ terms of $\mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_3$, respectively such that $
\left|\begin{array}{ccc}
a & 7 & 1 \\
2 b & 17 & 1 \\
c & 17 & 1
\end{array}\right|+70=0
$

If $a=29$, then the sum of the first 20 terms of an AP
whose first term is $c-a-b$ and the common difference is $\frac{d}{12}$, is equal to [JEE MAINS 2023]

Solution

$\begin{aligned} & \left|\begin{array}{ccc}A+6 d & 7 & 1 \\ 21(A+1+8 d) & 17 & 1 \\ A+2+16 d & 17 & 1\end{array}\right|+70=0 \\ & A=-7, d=6 \\ & \therefore \mathrm{c}-\mathrm{a}-\mathrm{b}=20 \\ & \therefore 5_{20}=495\end{aligned}$

Hence, the answer is 495.

Example 2:

Let $\alpha$ be a root of the equation $(a-c) x^2+(b-a) x+(c-b)=0$ where $\mathrm{a}, \mathrm{b}$, and c are distinct real numbers such that the matrix $\left[\begin{array}{ccc}\alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c\end{array}\right]$ is singular. Then, the value of $\frac{(a-c)^2}{(b-a)(c-b)}+\frac{(b-a)^2}{(a-c)(c-b)}+\frac{(c-b)^2}{(a-c)(b-a)}$ is

[JEE MAINS 2023]
Solution: $(a-c) x^2+(b-a) x+(c-b)=0$
$(a \neq c)$ $\mathrm{x}=1$ is one root \& other root is $\frac{\mathrm{c}-\mathrm{b}}{\mathrm{a}-\mathrm{c}}$
now $\left|\begin{array}{ccc}\alpha^2 & \alpha & 1 \\ 1 & 1 & 1 \\ a & b & c\end{array}\right|$ is singular

$
\Rightarrow\left|\begin{array}{ccc}
\alpha^2 & \alpha & 1 \\
1 & 1 & 1 \\
a & b & c
\end{array}\right|=0 \Rightarrow \alpha^2(c-b)-\alpha(c-a)+(b-a)=0
$
$\Rightarrow \alpha^2(c-b)+\alpha(a-c)+(b-a)=0
$

satisfied by $\alpha=1$ or $\alpha=\frac{b-a}{c-b}$
Now, if $\alpha=1$ then $\forall \mathrm{a} \neq \mathrm{b} \neq \mathrm{c}$

$
\begin{aligned}
& \sum \frac{(a-c)^2}{(b-a)(c-b)}=\frac{\sum(a-c)^3}{(a-b)(b-c)(c-a)} \\
& =\frac{3(a-b)(b-c)(c-a)}{(a-b)(b-c)(c-a)} \\
& =3
\end{aligned}
$

if $\left.A+B+C=0 \quad \Rightarrow \quad A^3+B^3+C^3=3 A B C\right]$

Hence, the answer is 3

Example 3: The probability that a randomly chosen $2 \times 2$ matrix with all the entries from the set of first 10 primes, is singular, is equal to :
[JEE MAINS 2022]
Solution: Let the matrix A be $\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$
If A is singular then $|A|=0$

$
\begin{aligned}
& \Rightarrow \mathrm{ad}-\mathrm{bc}=0 \\
& \Rightarrow \mathrm{ad}=\mathrm{bc}
\end{aligned}
$

Case I $: \mathrm{a}=\mathrm{d}$
a can take 10 values.
d can take the value which equals a
Now for ad $=$ be to hold

$
\mathrm{bc}=\mathrm{a}^2 \Rightarrow \mathrm{b}=\mathrm{a}, \mathrm{c}=\mathrm{a}(\mathrm{as} \mathrm{a} \text { is prime })
$

$\therefore$ only one option for b and c
Total such matrices $=10 \times 1 \times 1 \times 1=10$

Case II: $:$ ㅇd
In this case, can take 10 values and $d$ can take 9 values.
Now for $\mathrm{ad}=\mathrm{bc}, \quad(\mathrm{b}=\mathrm{a}$ and $\mathrm{c}=\mathrm{d})$
or $\quad(b=d$ and $c=a)$

$
\therefore \quad 10 \times 9 \times 2=180
$

$
\text { favourable }=10+180=190
$$
Total matrices $=10^4$

$
\text { Probability }=\frac{190}{10^4}=\frac{19}{10^3}
$

Hence, the answer is $\frac{19}{10^3

The value of $\left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right|_{\text {is : }}$
[MAINS 2021]
Solution: Put a $=0$, we get

$
\begin{aligned}
\Delta & =\left|\begin{array}{lll}
(a+1)(a+2) & a+2 & 1 \\
(a+2)(a+3) & a+3 & 1 \\
(a+3)(a+4) & a+4 & 1
\end{array}\right|=\left|\begin{array}{ccc}
2 & 2 & 1 \\
6 & 3 & 1 \\
12 & 4 & 1
\end{array}\right| \\
\Delta & =2(3-4)-2(6-12)+1(24-36)=-2
\end{aligned}
$

OR
Given matrix is

$
\begin{aligned}
\Delta & =\left|\begin{array}{lll}
(a+1)(a+2) & a+2 & 1 \\
(a+2)(a+3) & a+3 & 1 \\
(a+3)(a+4) & a+4 & 1
\end{array}\right| \\
\mathrm{R}_2 & \rightarrow \mathrm{R}_2-\mathrm{R}_1 \text { and } \mathrm{R}_3 \rightarrow \mathrm{R}_3-\mathrm{R}_1 \\
\Delta & =\left|\begin{array}{ccc}
(a+1)(a+2) & a+2 & 1 \\
(a+2)(a+3-a-1) & 1 & 0 \\
a^2+7 a+12-a^2-3 a-2 & 2 & 0
\end{array}\right| \\
& =\left|\begin{array}{ccc}
a^2+3 a+2 & a+2 & 1 \\
2(a+2) & 1 & 0 \\
4 a+10 & 2 & 0
\end{array}\right| \\
& =4(a+2)-4 a-10 \\
& =4 a+8-4 a-10=-2
\end{aligned}
$
Hence, the answer is -2

example 5: If the minimum and the maximum values of the function

$
\begin{aligned}
& f:\left[\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow R, \text { defined by } \\
& f(\theta)=\left|\begin{array}{ccc}
-\sin ^2 \theta & -1-\sin ^2 \theta & 1 \\
-\cos ^2 \theta & -1-\cos ^2 \theta & 1 \\
12 & 10 & -2
\end{array}\right|
\end{aligned}
$

are $m$ and $M$ respectively, then the ordered pair $(m, M)$ is equal to:
[JEE
[MAINS 2020]
Solution

$
\begin{aligned}
& \mathrm{C}_3 \rightarrow \mathrm{C}_3-\left(\mathrm{C}_1-\mathrm{C}_2\right) \\
& f(\theta)=\left|\begin{array}{ccc}
\sin ^2 \theta & -1-\sin ^2 \theta & 0 \\
-\cos ^2 \theta & -1-\cos ^2 \theta & 0 \\
12 & 10 & -4
\end{array}\right| \\
& =-4\left[\left(1+\cos ^2 \theta\right) \sin ^2 \theta-\cos ^2 \theta\left(1+\sin ^2 \theta\right)\right] \\
& =-4\left[\sin ^2 \theta+\cos ^2 \theta-\cos ^2 \theta-\cos ^2 \theta \sin ^2 \theta\right] \\
& f(\theta)=4 \cos 2 \theta \\
& \theta \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \\
& 2 \theta \in\left[\frac{\pi}{2}, \pi\right] \\
& (\theta) \in[-4,0] \\
& (\mathrm{m}, \mathrm{M})=(-4,0)
\end{aligned}
$

Hence, the answer is $(-4,0)$

Frequently Asked Questions (FAQs)

1. What are singular matrices?

A square matrix is called a singular matrix if its determinant is 0. Let's say A is a square matrix then it is singular if |A| = 0

2. What is determinant of matrices?

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

3. What is the determinant of matrix AB?

The product of the determinant of matrices AB is equal to the product of the determinant of individual matrices. det (AB) = det (A) det (B).

4. What is the difference between the determinant of a skew-symmetric matrix of even and odd order?

The determinant of the skew-symmetric matrix of odd order is zero. if A is the skew-symmetric matrix of odd order then |A| = 0. whereas, The determinant of the skew-symmetric matrix of even order is a perfect square. if A is a skew-symmetric matrix of even order then |A| is a perfect square.

5. What is det (A’)?

The determinant of the transpose of matrix A is equal to the determinant of matrix A. det (A’) = det A

6. How does a singular matrix affect the solution of linear equations?

When a matrix is singular, the system of linear equations it represents either has no solution or infinitely many solutions. This is because the rows of the matrix are linearly dependent, meaning there's not enough unique information to determine a single, unique solution.

7. How does matrix rank relate to singularity?

The rank of a matrix is the number of linearly independent rows or columns. A square matrix is singular if and only if its rank is less than its size (number of rows or columns). For a non-singular matrix, the rank is equal to its size.

8. What is the geometric interpretation of a singular matrix?

Geometrically, a singular matrix represents a linear transformation that "collapses" the space into a lower dimension. For example, a 2x2 singular matrix would transform a 2D plane into a line or a point, losing information in the process.

9. How does singularity affect matrix operations like multiplication?

Singularity affects matrix multiplication in several ways:

10. Can a singular matrix ever be diagonalizable?

Yes, a singular matrix can be diagonalizable, but not all singular matrices are diagonalizable. A singular matrix is diagonalizable if and only if its geometric multiplicity equals its algebraic multiplicity for the eigenvalue zero.

11. Why is the determinant of a singular matrix always zero?

The determinant of a matrix being zero indicates that the columns (or rows) of the matrix are linearly dependent. This means that one column can be expressed as a linear combination of the others, resulting in the matrix not having full rank and thus being singular.

12. How does adding or subtracting matrices affect singularity?

Adding or subtracting matrices can change their singularity. The sum or difference of two singular matrices isn't necessarily singular, and the sum or difference of two non-singular matrices isn't necessarily non-singular. It depends on how the operation affects the linear dependence of the columns or rows.

13. How do elementary row operations affect the singularity of a matrix?

Elementary row operations (multiplying a row by a non-zero scalar, adding a multiple of one row to another, or swapping rows) do not change the singularity of a matrix. If a matrix is singular, it will remain singular after these operations, and vice versa.

14. What is the relationship between eigenvalues and singular matrices?

A matrix is singular if and only if at least one of its eigenvalues is zero. This is because the determinant of a matrix is equal to the product of its eigenvalues, and a singular matrix has a determinant of zero.

15. What is the significance of the zero eigenvalue in a singular matrix?

A zero eigenvalue in a matrix indicates that the matrix is singular. It means there exists a non-zero vector x such that Ax = 0, which is the definition of the null space. The presence of a zero eigenvalue also implies that the determinant is zero.

16. What is a singular matrix?

A singular matrix is a square matrix that does not have an inverse. In other words, its determinant is equal to zero. This means that the matrix equations involving a singular matrix may have either no solution or infinitely many solutions.

17. How can you identify if a matrix is singular?

You can identify a singular matrix by calculating its determinant. If the determinant is zero, the matrix is singular. Another way is to check if the rank of the matrix is less than its size (number of rows or columns).

18. What are the implications of a matrix being singular in linear algebra?

When a matrix is singular, it means that the linear transformation represented by the matrix is not invertible. This has several implications:

19. Can a non-square matrix be singular?

The term "singular" is typically used only for square matrices. Non-square matrices are neither singular nor non-singular because they don't have determinants. However, we can talk about the rank of non-square matrices, which relates to their linear independence.

20. What's the difference between a singular matrix and a non-singular matrix?

The main differences are:

21. Can you explain the concept of pseudoinverse in relation to singular matrices?

The pseudoinverse, or Moore-Penrose inverse, is a generalization of the matrix inverse that can be applied to singular and non-square matrices. For singular matrices, the pseudoinverse provides a "best approximate solution" to systems of linear equations that have no exact solution or infinitely many solutions.

22. What is the relationship between singular matrices and matrix factorizations like LU decomposition?

Singular matrices pose challenges for many matrix factorizations. For example, the standard LU decomposition is not unique for singular matrices. Special techniques, like the complete LU factorization, are needed to handle singular matrices in these decompositions.

23. How does the concept of matrix similarity apply to singular matrices?

Two matrices A and B are similar if there exists an invertible matrix P such that B = P^(-1)AP. Similarity preserves eigenvalues, so if A is singular, any matrix similar to A is also singular. This concept is useful in studying the properties of singular matrices.

24. How does the concept of matrix similarity apply to singular matrices?

Similar matrices share many properties, including singularity. If A is singular and B is similar to A, then B is also singular. This is because similar matrices have the same eigenvalues, and a matrix is singular if and only if it has a zero eigenvalue.

25. How does the concept of matrix exponential apply to singular matrices?

The matrix exponential e^A is defined for all square matrices, including singular ones. However, for singular matrices, some properties of the exponential function that hold for non-singular matrices may not apply. For example, e^A may not be invertible if A is singular.

26. How does the concept of generalized inverse relate to singular matrices?

The generalized inverse, which includes the Moore-Penrose pseudoinverse, provides a way to "invert" singular matrices. It's useful in solving least squares problems and finding minimum norm solutions to underdetermined systems.

27. How does the concept of matrix factorization techniques like SVD apply to singular matrices?

Singular Value Decomposition (SVD) is particularly useful for singular matrices. It provides insights into the null space and range of the matrix, and forms the basis for techniques like principal component analysis and pseudoinverse computation.

28. How does the concept of matrix perturbation theory relate to singular matrices?

Matrix perturbation theory studies how small changes in a matrix affect its properties. For nearly singular matrices, small perturbations can lead to large changes in properties like eigenvalues and eigenvectors, which is important in numerical linear algebra.

29. Can you explain the concept of null space in relation to singular matrices?

The null space of a matrix A is the set of all vectors x such that Ax = 0. For a singular matrix, the null space is non-trivial (contains non-zero vectors). This means there are non-zero solutions to the equation Ax = 0, which is directly related to the matrix's singularity.

30. What is the connection between linear dependence and singular matrices?

A matrix is singular if and only if its columns (or rows) are linearly dependent. This means that at least one column can be expressed as a linear combination of the others, indicating that the matrix doesn't contain enough unique information to be invertible.

31. How does singularity relate to the concept of matrix conditioning?

Matrix conditioning measures how sensitive a linear system is to errors in the input data. Singular matrices are considered "infinitely ill-conditioned" because small changes in the input can lead to arbitrarily large changes in the output, making computations unstable.

32. How does the trace of a matrix relate to its singularity?

The trace of a matrix (sum of diagonal elements) doesn't directly determine singularity. However, if a matrix has a non-zero trace, it cannot be nilpotent (a special type of singular matrix where some power of the matrix equals the zero matrix). But a non-zero trace doesn't guarantee non-singularity.

33. What is the relationship between singular matrices and linear transformations?

A singular matrix represents a linear transformation that is not one-to-one (injective) or onto (surjective). It maps vectors from the domain to a lower-dimensional subspace of the codomain, effectively "losing" information in the process.

34. What is the connection between singular matrices and systems of homogeneous linear equations?

A system of homogeneous linear equations Ax = 0 has non-trivial solutions (solutions other than the zero vector) if and only if A is singular. This is because the existence of non-trivial solutions implies that the columns of A are linearly dependent.

35. How does the concept of linear independence relate to singular matrices?

A matrix is singular if and only if its columns (or rows) are linearly dependent. Linear independence means that no column can be expressed as a linear combination of the others. In a singular matrix, at least one column can be expressed this way, indicating redundancy in the information contained in the matrix.

36. What is the significance of the reduced row echelon form for singular matrices?

The reduced row echelon form of a singular matrix will always have at least one row of all zeros. This form clearly shows the linear dependence of the rows and helps in determining the rank of the matrix, which is less than its size for singular matrices.

37. How does singularity affect the solution space of a system of linear equations?

For a singular coefficient matrix, the solution space of the system Ax = b is either empty (no solution) or infinite (infinitely many solutions). If solutions exist, they form an affine subspace of dimension equal to the nullity of the matrix (dimension of the null space).

38. Can you explain the concept of generalized eigenvectors in relation to singular matrices?

Generalized eigenvectors are particularly important for singular matrices that are not diagonalizable. They help in forming a Jordan canonical form of the matrix, which is useful for understanding the matrix's structure and behavior, especially when it has repeated eigenvalues (including zero).

39. How does the determinant function behave near singular matrices?

The determinant function is continuous, but it behaves in an interesting way near singular matrices. As a matrix approaches singularity, its determinant approaches zero. This can lead to numerical instability in computations involving nearly singular matrices.

40. What is the significance of the characteristic polynomial for singular matrices?

For a singular matrix, the characteristic polynomial always has zero as a root. This is because the characteristic polynomial is det(λI - A), and for a singular matrix A, det(A) = 0, so λ = 0 is always a solution to the characteristic equation.

41. What is the relationship between singular matrices and projections?

Projection matrices are always singular (except for the identity matrix). This is because projections "collapse" a space onto a subspace, which is inherently a non-invertible operation. Understanding singular matrices is crucial for working with projections in linear algebra.

42. How does the concept of matrix norm relate to singular matrices?

For singular matrices, some matrix norms can be zero or undefined. For example, the induced 2-norm of a matrix (its largest singular value) is zero only for the zero matrix, but other norms like the Frobenius norm can be non-zero for singular matrices.

43. What is the connection between singular matrices and linear regression?

In linear regression, singularity of the design matrix leads to the problem of multicollinearity. This occurs when the predictors are linearly dependent, making it impossible to uniquely estimate the regression coefficients. Techniques like ridge regression are used to handle such cases.

44. How does the concept of matrix condition number relate to singular matrices?

The condition number of a matrix is the ratio of its largest to smallest singular value. For singular matrices, the smallest singular value is zero, making the condition number infinite. This indicates extreme sensitivity to perturbations and numerical instability in computations.

45. What is the significance of the Jordan canonical form for singular matrices?

The Jordan canonical form is particularly useful for understanding the structure of singular matrices. It reveals the dimensions of the eigenspaces and generalized eigenspaces, which is crucial for analyzing the behavior of the matrix in applications like solving differential equations.

46. How does the concept of matrix pencils relate to singular matrices?

A matrix pencil A - λB is singular if det(A - λB) = 0 for all λ. This concept generalizes the idea of matrix singularity and is important in areas like control theory and differential-algebraic equations.

47. What is the relationship between singular matrices and Markov chains?

In Markov chains, a singular transition matrix indicates a reducible chain, where some states are not accessible from others. This has important implications for the long-term behavior of the system and the existence of stationary distributions.

48. What is the significance of the nullity of a matrix in relation to its singularity?

The nullity of a matrix is the dimension of its null space. For a square matrix, non-zero nullity implies singularity. The nullity plus the rank equals the size of the matrix, known as the rank-nullity theorem.

49. What is the relationship between singular matrices and linear differential equations?

In systems of linear differential equations dx/dt = Ax, singularity of A affects the nature of solutions. If A is singular, the system may have non-unique solutions or solutions that grow unboundedly, depending on the Jordan structure of A.

50. How does the concept of matrix power series relate to singular matrices?

For a singular matrix A, the power series I + A + A^2 + ... may not converge. This is related to the spectral radius of A. Understanding the behavior of such series for singular matrices is important in areas like Markov chains and dynamical systems.

51. What is the significance of the Kronecker product in relation to singular matrices?

The Kronecker product of two matrices is singular if either of the original matrices is singular. This property is useful in studying tensor products of linear transformations and in areas like quantum computing.

52. How does the concept of matrix logarithm apply to singular matrices?

The matrix logarithm is not defined for singular matrices. This is because the logarithm of zero is undefined, and singular matrices effectively have zero as an eigenvalue. This limitation is important in applications involving matrix functions.

53. What is the relationship between singular matrices and the concept of matrix rank?

A square matrix is singular if and only if its rank is less than its size. The rank-nullity theorem states that for an m × n matrix A, rank(A) + nullity(A) = n. For singular matrices, this means the nullity is always positive.

54. What is the significance of the characteristic subspaces for singular matrices?

For a singular matrix, the characteristic subspace corresponding to the eigenvalue zero (the null space) is non-trivial. Understanding these subspaces is crucial for analyzing the action of the matrix and solving related linear systems.

55. What is the relationship between singular matrices and the concept of matrix decomposition?

Various matrix decompositions behave differently for singular matrices. For example, LU decomposition may not be unique, QR decomposition with column pivoting can reveal rank deficiency, and SVD provides a robust way to analyze the structure of singular matrices.