What are some properties of the Symmetric and Skew Symmetric Matrix?

i) If A is a square matrix, then AA’ and A’A are symmetric matrices ii) If A is a symmetric matrix, then -A, kA, A’, A n , B’AB are also symmetric matrices where n ∈ N, k ∈ R, and B is a square matrix of order same as matrix A. iii) If A is a skew-symmetric matrix then A 2n is a symmetric matrix for n ∈ N. A 2n+1 is a skew-symmetric matrix for n ∈ N kA is also a skew-symmetric matrix, where k ∈ R B’AB is also a skew-symmetric matrix where B is a square matrix of order same as matrix A iv) If A and B are symmetric matrices then: A ± B, AB+BA are symmetric matrices. AB - BA is a skew-symmetric matrix. v) If A and B are skew-symmetric matrices then: A ± B, AB - BA are skew-symmetric matrices. AB + BA is a symmetric matrix.

Does conjugate exist only for square matrices?

No, the Conjugate of the matrix can be applied to any matrix whether it is square matrix or rectangular matrix.

What is the relationship between a skew-symmetric matrix and its determinant?

For a skew-symmetric matrix of odd order (size), its determinant is always zero. For even-ordered skew-symmetric matrices, the determinant is always non-negative and is the square of a polynomial in the matrix entries, known as the Pfaffian.

How do symmetric and skew-symmetric matrices relate to the singular value decomposition (SVD)?

For a symmetric matrix, the singular value decomposition is equivalent to the spectral decomposition, and the singular values are the absolute values of the eigenvalues. For a skew-symmetric matrix, the singular values come in pairs (except for possibly one zero singular value in odd dimensions).

How do symmetric and skew-symmetric matrices relate to graph theory?

In graph theory, symmetric matrices often represent undirected graphs (adjacency matrices), while skew-symmetric matrices can represent directed graphs or tournaments. The properties of these matrices provide insights into graph characteristics such as connectivity, cycles, and matchings.

How can you use the properties of skew-symmetric matrices in solving differential equations?

Skew-symmetric matrices are useful in solving certain types of differential equations, particularly those involving rotations or angular velocities. The exponential of a skew-symmetric matrix represents a rotation, which is used in solving systems of linear differential equations with constant coefficients.

How do eigenvalues of a skew-symmetric matrix differ from those of a symmetric matrix?

While eigenvalues of symmetric matrices are always real, eigenvalues of skew-symmetric matrices are either zero or purely imaginary (i.e., complex numbers with zero real part). This is because the characteristic polynomial of a skew-symmetric matrix has only even powers if the matrix has even order, and only odd powers if it has odd order.

What are the key properties of eigenvalues in symmetric matrices?

Eigenvalues of symmetric matrices have several important properties: 1) They are always real numbers, not complex. 2) The eigenvectors corresponding to distinct eigenvalues are orthogonal. 3) A symmetric matrix is positive definite if and only if all its eigenvalues are positive.

What is the significance of positive definite symmetric matrices?

Positive definite symmetric matrices are particularly important in various fields. They have all positive eigenvalues, represent convex quadratic forms, and arise in many applications such as covariance matrices in statistics, stiffness matrices in structural engineering, and Hessian matrices in optimization problems.

How can you decompose a symmetric matrix?

A symmetric matrix can be decomposed in several ways: 1) Spectral decomposition: A = QΛQT, where Q is orthogonal and Λ is diagonal. 2) Cholesky decomposition: If the matrix is positive definite, it can be written as A = LLT, where L is lower triangular. These decompositions are useful in various computational and theoretical contexts.

How does a skew-symmetric matrix differ from a symmetric matrix?

A skew-symmetric matrix is a square matrix whose transpose is equal to its negative. This means that for any element aij in the matrix, aij = -aji. Additionally, all elements on the main diagonal of a skew-symmetric matrix are zero.

Can a matrix be both symmetric and skew-symmetric?

Yes, a matrix can be both symmetric and skew-symmetric, but only if it's a zero matrix. This is because the conditions for both types require aij = aji (symmetric) and aij = -aji (skew-symmetric), which is only true when all elements are zero.

How can you determine if a matrix is symmetric without calculating its transpose?

To determine if a matrix is symmetric without explicitly calculating its transpose, check if the matrix is square and then verify that each element aij is equal to aji for all i and j. In other words, check if the matrix is "mirror-imaged" across its main diagonal.

How does matrix addition affect symmetry or skew-symmetry?

The sum of two symmetric matrices is always symmetric, and the sum of two skew-symmetric matrices is always skew-symmetric. However, adding a symmetric matrix to a skew-symmetric matrix results in neither a symmetric nor a skew-symmetric matrix, unless one of the matrices is zero.

What is the significance of the main diagonal in symmetric and skew-symmetric matrices?

In a symmetric matrix, the main diagonal can contain any real numbers. In a skew-symmetric matrix, all elements on the main diagonal must be zero. This is because in a skew-symmetric matrix, aij = -aji, and for diagonal elements where i = j, this implies aii = -aii, which is only true when aii = 0.

Can you have a non-square symmetric or skew-symmetric matrix?

No, symmetric and skew-symmetric matrices must be square. The definitions of these matrices rely on the relationship between aij and aji, which only exists in square matrices where the number of rows equals the number of columns.

What is the relationship between skew-symmetric matrices and cross products?

In three-dimensional space, every skew-symmetric 3x3 matrix corresponds to a unique vector, and vice versa. This relationship is used to represent cross products as matrix multiplications, which is particularly useful in physics and computer graphics.

What is the trace of a skew-symmetric matrix and why?

The trace of a skew-symmetric matrix is always zero. This is because all elements on the main diagonal of a skew-symmetric matrix are zero, and the trace is defined as the sum of these diagonal elements.

How do symmetric and skew-symmetric matrices relate to linear transformations?

Symmetric matrices represent self-adjoint linear transformations, which preserve inner products. Skew-symmetric matrices represent anti-symmetric transformations. In physics, symmetric matrices often represent conservative forces, while skew-symmetric matrices can represent rotations or angular velocities.

What is the connection between Hermitian matrices and real symmetric matrices?

Hermitian matrices are the complex analog of real symmetric matrices. A real symmetric matrix is a special case of a Hermitian matrix where all entries are real. Many properties of symmetric matrices extend to Hermitian matrices in the complex domain.

What is a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its transpose. In other words, if you flip a symmetric matrix over its main diagonal (from top-left to bottom-right), it remains unchanged. For any element aij in the matrix, aij = aji.

How can you construct a symmetric matrix from any square matrix?

You can construct a symmetric matrix from any square matrix A by computing (A + AT)/2. This operation is called "symmetrization" and always results in a symmetric matrix, regardless of whether the original matrix was symmetric or not.

What is the connection between symmetric matrices and quadratic forms?

Every quadratic form can be represented by a symmetric matrix, and every symmetric matrix defines a quadratic form. This connection is crucial in optimization problems, where the nature of critical points (minima, maxima, or saddle points) is determined by the properties of the associated symmetric matrix (positive definite, negative definite, or indefinite).

How does matrix multiplication interact with symmetry?

The product of two symmetric matrices is not necessarily symmetric. However, if A is a symmetric matrix, then AAT (A multiplied by its transpose) is always symmetric. Additionally, if A is any square matrix, both ATA and AAT are always symmetric.

What is the connection between orthogonal matrices and symmetric matrices?

An orthogonal matrix Q multiplied by a symmetric matrix S and then by the transpose of Q (i.e., QSQT) results in another symmetric matrix. This property is fundamental in the spectral theorem and diagonalization of symmetric matrices.

How does the inverse of a symmetric matrix behave?

If a symmetric matrix is invertible, its inverse is also symmetric. This property is useful in many applications, including solving systems of linear equations and in optimization algorithms.

What is the role of symmetric matrices in principal component analysis (PCA)?

In PCA, the covariance or correlation matrix, which is always symmetric, is analyzed. The eigenvectors of this matrix correspond to the principal components, and the eigenvalues represent the amount of variance explained by each component. This illustrates the importance of symmetric matrices in data analysis and dimensionality reduction.

How can you use the properties of symmetric matrices to solve systems of linear equations efficiently?

When solving Ax = b where A is symmetric, methods like the conjugate gradient algorithm can be used, which are more efficient than general methods for non-symmetric matrices. These methods exploit the symmetry to reduce computational complexity and improve numerical stability.

What is the significance of the eigenvalue distribution in large random symmetric matrices?

The study of eigenvalue distributions in large random symmetric matrices is important in various fields, including physics and data science. The semicircle law, which describes the distribution of eigenvalues in certain random symmetric matrices as their size approaches infinity, has applications in quantum chaos and complex systems.

What is the connection between symmetric matrices and optimization problems?

Symmetric matrices play a crucial role in optimization, particularly in quadratic programming. The Hessian matrix of a twice-differentiable function is symmetric, and its properties (positive definite, negative definite, or indefinite) determine the nature of critical points in optimization problems.

How do symmetric and skew-symmetric matrices behave under similarity transformations?

Under a similarity transformation S^(-1)AS, a symmetric matrix remains symmetric, and a skew-symmetric matrix remains skew-symmetric. This invariance is important in studying the properties of these matrices that are independent of the choice of basis.

What is the relationship between symmetric matrices and inner product spaces?

In an inner product space, every symmetric bilinear form can be represented by a unique symmetric matrix with respect to a given basis. This connection is fundamental in the study of quadratic forms and metric spaces.

What is the significance of the Cayley transform for skew-symmetric matrices?

The Cayley transform provides a bijection between skew-symmetric matrices and special orthogonal matrices (except those with eigenvalue -1). This transform is useful in numerical algorithms for computing matrix exponentials and in the study of Lie groups and algebras.

How do symmetric and skew-symmetric matrices relate to the Jordan canonical form?

For symmetric matrices, the Jordan canonical form is always diagonal, which is equivalent to the spectral decomposition. For skew-symmetric matrices, the Jordan blocks are either 1x1 (corresponding to zero eigenvalues) or 2x2 (corresponding to pairs of imaginary eigenvalues).

How can you use the properties of symmetric matrices to speed up matrix computations?

The symmetry of a matrix can be exploited to reduce storage requirements and computational complexity. For example, only the upper or lower triangular part of a symmetric matrix needs to be stored, and many algorithms can be optimized to take advantage of this structure, leading to significant performance improvements.

What is the connection between symmetric matrices and distance metrics?

Symmetric positive definite matrices can define distance metrics in vector spaces. The Mahalanobis distance, which is widely used in statistics and machine learning, is defined using a symmetric positive definite matrix. This connection illustrates the importance of symmetric matrices in defining and working with generalized notions of distance.

What is the significance of the commutator of two matrices in relation to symmetry?

The commutator of two matrices A and B, defined as [A,B] = AB - BA, is always skew-symmetric if A and B are symmetric. This property is important in quantum mechanics, where commutators of Hermitian operators (complex analogs of real symmetric matrices) represent observables.

How can you use the properties of symmetric matrices in spectral clustering algorithms?

In spectral clustering, the Laplacian matrix of a graph, which is symmetric, is used to perform dimensionality reduction before clustering. The eigenvectors of this matrix provide a spectral embedding of the data, illustrating how the properties of symmetric matrices can be leveraged for data analysis and clustering tasks.

What is the relationship between orthogonal diagonalization and symmetric matrices?

Every real symmetric matrix is orthogonally diagonalizable, meaning it can be written as A = QΛQT, where Q is orthogonal and Λ is diagonal. This property, known as the spectral theorem, is fundamental in many applications and theoretical developments involving symmetric matrices.

Symmetric Matrix & Skew Symmetric Matrix

Maths
Matrices
Symmetric Matrix & Skew Symmetric Matrix

Symmetric Matrix & Skew Symmetric Matrix

Q: How do symmetric and skew-symmetric matrices behave under matrix exponentiation?

The exponential of a symmetric matrix is always positive definite. The exponential of a skew-symmetric matrix is always orthogonal. These properties are important in the study of matrix Lie groups and differential equations.

Q: What is the relationship between skew-symmetric matrices and Lie algebras?

Skew-symmetric matrices form the Lie algebra of the special orthogonal group SO(n). This connection is fundamental in the study of continuous symmetries and has applications in physics, particularly in quantum mechanics and particle physics.

Komal MiglaniUpdated on 02 Jul 2025, 06:33 PM IST

A matrix is a rectangular arrangement of symbols along rows and columns that might be real or complex numbers. A system of m x n symbols arranged in a rectangular formation along m rows and n columns and bonded by the brackets [ ] is called an m by n matrix (which is written as m x n matrix). Symmetric matrices are helpful in solving problems in areas like statistics, physics, and optimization. On the other hand, skew-symmetric matrices are used in fields like mechanics and electromagnetism.

This Story also Contains

Symmetric Matrix
Skew-symmetric matrix
Properties of Symmetric and Skew-symmetric Matrices:
Determinant of Skew Symmetric Matrix
Eigenvalues of Skew Symmetric Matrix
Summary

In this article, we will cover the concept of Symmetric and Skew Symmetric 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. Over the last ten years of the JEE Main Exam (from 2013 to 2023), a total of 6 questions have been asked on this concept, including one in 2019, two in 2021, two in 2022, and one in 2023.

Symmetric Matrix

A square matrix $A=\left[a_{i j}\right]_{n \times n}$ is said to be symmetric if $A^{\prime}=A$, i.e., $a_{i j}=a_{j i} \forall i, j$
$
\mathrm{A}=\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right] \text { then } \mathrm{A}^{\prime}=\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right]
$

Clearly, $\mathrm{A}=\mathrm{A}^{\prime}$, hence $\mathrm{A}$ is a symmetric matrix

Skew-symmetric matrix

A square matrix $A=\left[a_{i j}\right]_{m \times n}$ is said to be skew-symmetric if $A^{\prime}=-A$
$
\text { i.e. } \mathrm{A}^{\prime}=-\mathrm{A} \text {, i.e., } \mathrm{a}_{\mathrm{ij}}=-\mathrm{a}_{\mathrm{ji}} \forall \mathrm{i}, \mathrm{j}
$

Now if we put $\mathrm{i}=\mathrm{j}$, we have
$
\begin{aligned}
& \mathrm{a}_{\mathrm{ii}}=-\mathrm{a}_{\mathrm{ii}}, \\
& \therefore 2 \mathrm{a}_{\mathrm{ii}}=0 \Rightarrow \mathrm{a}_{\mathrm{ii}}=0 \forall \mathrm{i}^{\prime} \mathrm{s}
\end{aligned}
$

That means all the diagonal elements of a skew-symmetric matrix are 0.
e.g. $\mathrm{A}=\left[\begin{array}{ccc}0 & h & g \\ -h & 0 & f \\ -g & -f & 0\end{array}\right]$, then $\mathrm{A}^{\prime}=\left[\begin{array}{ccc}0 & -h & -g \\ h & 0 & -f \\ g & f & 0\end{array}\right]=-\mathrm{A}$

Properties of Symmetric and Skew-symmetric Matrices:

i) If A is a square matrix, then AA’ and A’A are symmetric matrices

ii) If A is a symmetric matrix, then -A, kA, A’, Aⁿ, B’AB are also symmetric matrices where n ∈ N, k ∈ R, and B is a square matrix of order same as matrix A.

iii) If A is a skew-symmetric matrix then

A²ⁿ is a symmetric matrix for n ∈ N.
A²ⁿ⁺¹ is a skew-symmetric matrix for n ∈ N
kA is also a skew-symmetric matrix, where k ∈ R
B’AB is also a skew-symmetric matrix where B is a square matrix of order same as matrix A

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iv) If A and B are symmetric matrices then:

A ± B, AB+BA are symmetric matrices.
AB - BA is a skew-symmetric matrix.

v) If A and B are skew-symmetric matrices then:

A ± B, AB - BA are skew-symmetric matrices.
AB + BA is a symmetric matrix.

Determinant of Skew Symmetric Matrix

A Skew-Symmetric matrix is square in shape, and its determinant satisfies the requirement that is covered in the section that follows. In the event that our matrix is skew-symmetric,

$\operatorname{Det}\left(A^{\top}\right)=\operatorname{det}(-A)=(-1)^n \operatorname{det}(A)$

Furthermore, each odd-order skew-symmetric matrix is a singular matrix, meaning that its determinant is 0, meaning that it does not exist.

Eigenvalues of Skew Symmetric Matrix

A skew-symmetric matrix has zero eigenvalues. Although the matrix may have non-real eigenvalues, it is a real matrix. Additionally, it is simple to represent every square matrix as the unique sum of a symmetric and a skew-symmetric matrix.

Summary

Owing to their unique structure, skew-symmetric matrices can be computed more efficiently than general matrices, which can provide computational advantages in some simulations and algorithms because symmetric matrices frequently need fewer computations than ordinary matrices, making use of their symmetry can result in computational advantages. Symmetric matrices exhibit symmetry across their main diagonal, ensuring real eigenvalues and orthogonal eigenvectors. This symmetry supports efficient algorithms in numerical computations and optimization problems.

Recommend Video:

Solved Examples Based on Symmetric and Skew Symmetric matrix

$\text { Example 1: Let } A=\left[\begin{array}{rr}
0 & -2 \\
2 & 0
\end{array}\right] \text {. If } \mathrm{M} \text { and } \mathrm{N} \text { are two matrices given by } M=\sum_{k=1}^{10} A^{2 k} \text { and } N=\sum_{k=1}^{10} A^{2 k-1} \text {. Then } \mathrm{MN}^2 \text { is: } $

Solution:

$
\mathrm{A}^2=\left[\begin{array}{cc}
0 & -2 \\
2 & 0
\end{array}\right]\left[\begin{array}{cc}
0 & -2 \\
2 & 0
\end{array}\right]=\left[\begin{array}{cc}
-4 & 0 \\
0 & -4
\end{array}\right]=-4 \mathrm{I}: \text { symmetric }
$
$\& \mathrm{~A}^3=-4 \mathrm{~A}$ (Skew Symmetric)
$
\begin{aligned}
& \Rightarrow \mathrm{M}=\sum_{\mathrm{k}=1}^{10} \mathrm{~A}^{2 \mathrm{k}}=\left[(-4)+(-4)^2+(-4)^3+\cdots+(-4)^{10}\right] \mathrm{I} \\
& =-4 \lambda \mathrm{I} \text { is Symmetric } \\
& \Rightarrow \mathrm{N}=\sum_{\mathrm{k}=1}^{10} \mathrm{~A}^{2 \mathrm{k}-1}=\mathrm{A}\left[1+(-4)+(-4)^3+\cdots+(-4)^9\right] \mathrm{I} \\
& =\lambda \mathrm{A} \text { is Skew Symmetric } \\
&
\end{aligned}
$
where $\lambda=\left\{1+(-4)+(-4)^3+\cdots+(-4)^9\right\}$
$
M N^2=-4 \lambda^3 A^2
$
$\Rightarrow \mathrm{MN}^2$ is Symmetric matrix

Example 2: If $\mathrm{A}$ is a symmetric matrix and $\mathrm{B}$ is a skew-symmetric matrix such that $A+B=\left[\begin{array}{cc}2 & 3 \\ 5 & -1\end{array}\right]$, then $\mathrm{AB}$ is equal to :
Solution:
Symmetric matrix - If $A=\left[a_{i j}\right]$ and $a_{i j}=a_{j i}$ for all $i$ and $j$

Skew symmetric matrix - If $A=\left[a_{i j}\right]$ and $a_{i j}=-a_{j i}$ for all $i$ and $j$
$
\left[\begin{array}{ccc}
0 & 2 & -1 \\
-2 & 0 & -4 \\
1 & 4 & 0
\end{array}\right]
$

A is a symmetric matrix
$
A^{\prime}=A
$
$B$ is skew-Symmetrix
$
\begin{aligned}
& B^{\prime}=-B \\
& A+B=\left[\begin{array}{cc}
2 & 3 \\
5 & -1
\end{array}\right] \cdots \cdots(1) \\
& A^{\prime}+B^{\prime}=\left[\begin{array}{cc}
2 & 5 \\
3 & -1
\end{array}\right] \\
& A-B=\left[\begin{array}{cc}
2 & 5 \\
3 & -1
\end{array}\right] \cdots \cdots \cdot(2)
\end{aligned}
$

(1) + (2)

$\begin{aligned} & A=\left[\begin{array}{cc}2 & 4 \\ 4 & -1\end{array}\right], B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right] \\ & A B=\left[\begin{array}{cc}4 & -2 \\ -1 & -4\end{array}\right]\end{aligned}$

Example 3: Let $\mathrm{A}$ be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^2$ is 1, then the possible number of such matrices is:
Solution:
$
\begin{aligned}
& A=\left(\begin{array}{ll}
a & b \\
b & c
\end{array}\right), \quad a, b, c \in I \\
& A^2=\left(\begin{array}{ll}
a & b \\
b & c
\end{array}\right)\left(\begin{array}{ll}
a & b \\
b & c
\end{array}\right)=\left(\begin{array}{cc}
a^2+b^2 & b(a+c) \\
b(a+c) & b^2+c^2
\end{array}\right)
\end{aligned}
$

A sum of all the diagonal elements of
$
A^2=a^2+2 b^2+c^2
$

Given $a^2+2 b^2+c^2=1, a, b, c \in I$
$
b=0 \& a^2+c^2=1
$

Case-1 : $\mathrm{a}=0 \Rightarrow \mathrm{c}= \pm 1 \quad(2$-matrices )
Case-2 : $c=0 \Rightarrow a= \pm 1 \quad$ (2-matrices)
Total $=4$ matrices

Hence, the possible number of such matrices is 4

Example 4: Let $A=\left[\begin{array}{ll}2 & 3 \\ a & 0\end{array}\right], a \in \mathbf{R}$ be written as $P+Q_{\text {where }} \mathrm{P}$ is a symmetric matrix and $\mathrm{Q}$ is a skew-symmetric matrix. If det(Q) $=9$, then the modulus of the sum of all possible values of a determinant of $\mathrm{P}$ is equal to?

Solution:

Using the property of matrices,

$
A=P+Q
$
$A$ can be written sum of a symmetric and a skew-symmetric matrix, Where $P=\frac{1}{2}\left(A+A^{\prime}\right)$ and $Q=\frac{1}{2}\left(A-A^{\prime}\right)$
$
\begin{aligned}
\therefore \quad Q & =\frac{1}{2}\left(A-A^{\prime}\right) \\
& =\frac{1}{2}\left(\left[\begin{array}{ll}
2 & 3 \\
a & 0
\end{array}\right]-\left[\begin{array}{ll}
2 & a \\
3 & 0
\end{array}\right]\right) \\
& =\frac{1}{2}\left[\begin{array}{cc}
0 & 3-a \\
a-3 & 0
\end{array}\right] \\
& =\left[\begin{array}{cc}
0 & \frac{3-a}{2} \\
\frac{a-3}{2} & 0
\end{array}\right]
\end{aligned}
$

Given $|Q|=9$
$
\begin{aligned}
& \Rightarrow 0-\left(\frac{a-3}{2}\right)\left(\frac{3-a}{2}\right)=9 \\
& \Rightarrow(a-3)^2=36 \\
& \Rightarrow a-3=6 \text { or } a-3=-6 \\
& \Rightarrow a=9 \text { or } a=-3 .
\end{aligned}
$

So,

$
\begin{aligned}
& A=\left[\begin{array}{ll}
2 & 3 \\
9 & 0
\end{array}\right] \text { or } A=\left[\begin{array}{cc}
2 & 3 \\
-3 & 0
\end{array}\right] \\
& P=\frac{1}{2}\left(\left[\begin{array}{ll}
2 & 3 \\
9 & 0
\end{array}\right]+\left[\begin{array}{ll}
2 & 9 \\
3 & 0
\end{array}\right]\right) \text { or } P=\frac{1}{2}\left(\left[\begin{array}{cc}
2 & 3 \\
-3 & 0
\end{array}\right]+\left[\begin{array}{cc}
2 & -3 \\
3 & 0
\end{array}\right]\right) \\
& P=\left[\begin{array}{ll}
2 & 6 \\
6 & 0
\end{array}\right] \quad \text { or } P=\left[\begin{array}{ll}
2 & 0 \\
0 & 0
\end{array}\right] \\
& \Rightarrow|P|=-36 \text { or }|P|=0
\end{aligned}
$

Sum of possible $|P|=-36+0=-36$.
Mod of this value $=36$

Hence, the modulus of the sum of all possible values of a determinant of P is 36.

Example 5: Let $A$ be a symmetric matrix such that $|A|=2$ and $\left[\begin{array}{cc}2 & 1 \\ 3 & \frac{3}{2}\end{array}\right] A-\left[\begin{array}{cc}1 & 2 \\ \alpha & \beta\end{array}\right]$. If the sum of the diagonal elements of $A$ is $s$, then $\frac{\beta s}{\alpha^2}$ is equal to

Solution:

A symmetric matrix such that $
|A|=2 \text { and }\left[\begin{array}{ll}
2 & 1 \\
3 & 3 / 2
\end{array}\right] \mathrm{A}=\left[\begin{array}{ll}
1 & 2 \\
\alpha & \beta
\end{array}\right]
$
$
\begin{aligned}
& A=\left[\begin{array}{ll}
a & b \\
b & d
\end{array}\right] \quad|A|=a d-b^2=2 \\
& {\left[\begin{array}{cc}
2 & 1 \\
3 & 3 / 2
\end{array}\right]\left[\begin{array}{ll}
\mathrm{a} & \mathrm{b} \\
\mathrm{b} & \mathrm{d}
\end{array}\right]=\left[\begin{array}{ll}
1 & 2 \\
\alpha & \beta
\end{array}\right]} \\
& {\left[\begin{array}{cc}
2 a+b & 2 b+d \\
3 a+3 / 2 b & 3 b+3 / 2 d
\end{array}\right]=\left[\begin{array}{ll}
1 & 2 \\
\alpha & \beta
\end{array}\right]} \\
& 2 \mathrm{a}+\mathrm{b}=1 \quad 2 \mathrm{~b}+\mathrm{d}=2 \\
& \mathrm{~b}=1-2 \mathrm{a} \quad \mathrm{d}=2-2 \mathrm{~b} \\
& =2-2(1-2 a) \\
& =2-2+4 a \\
& a d-b^2=2 \\
& a .4 a-(1-2 a)^2=2 \quad \text { Now } \alpha=3 a+\frac{3}{2} b \\
&
\end{aligned}
$

$\begin{aligned} & \Rightarrow 4 \mathrm{a}^2-1-4 \mathrm{a}^2+4 \mathrm{a}=2 \quad=\frac{9}{4}+\frac{3}{2} \cdot\left(\frac{-1}{2}\right) \\ & 4 \mathrm{a}=3, \quad=\frac{9-3}{4}=\frac{6}{4}=\frac{3}{2} \\ & \mathrm{a}=\frac{3}{4} \\ & b=1-2 \times \frac{3}{4} \quad \beta=3 b+\frac{3}{2} d \\ & =\frac{-1}{2} \quad 3 \times\left(\frac{-1}{2}\right)+\frac{3}{2} \times 3 \\ & \mathrm{~d}=4 \times \frac{3}{4}=3 \quad \frac{-3+9}{2}=3 \\ & \mathrm{~A}=\left[\begin{array}{cc}3 / 4 & -1 / 2 \\ -1 / 2 & 3\end{array}\right] \mathrm{s}=\frac{3}{4}+3=\frac{15}{4} \\ & \frac{\mathrm{Bs}}{\alpha^2}=\frac{3 \times \frac{15}{4}}{\frac{9}{4}}=5 \\ & \end{aligned}$

Hence, the answer is the 5.

Frequently Asked Questions (FAQs)

Q: How do symmetric and skew-symmetric matrices relate to the theory of Lie groups and Lie algebras?

Symmetric matrices are related to the symmetric spaces associated with Lie groups, while skew-symmetric matrices form the Lie algebra of the special orthogonal group. This connection is fundamental in the study of continuous symmetries and has far-reaching applications in theoretical physics and differential geometry.

Q: What is the significance of the Toeplitz structure in symmetric matrices?

A Toeplitz matrix is constant along diagonals. When a Toeplitz matrix is also symmetric, it has a special structure where each diagonal is constant and the matrix is fully determined by its first row (or column). This structure arises in signal processing and time series analysis and allows for efficient algorithms for matrix operations.

Q: How can you use the properties of symmetric matrices in solving differential equations numerically?

In numerical methods for solving differential equations, such as the finite element method, symmetric matrices often arise in the discretization of elliptic partial differential equations. The symmetry can be exploited to use more efficient solvers and to ensure certain properties of the numerical solution, such as energy conservation.

Q: What is the role of symmetric matrices in principal axis theorems?

The principal axis theorem states that any quadratic form can be transformed into a sum of squares by an orthogonal change of variables. This theorem is intimately connected with the diagonalization of symmetric matrices and has applications in physics, engineering, and data analysis.

Q: How do symmetric and skew-symmetric matrices behave under the Kronecker product?

The Kronecker product of two symmetric matrices is symmetric, and the Kronecker product of two skew-symmetric matrices is symmetric. However, the Kronecker product of a symmetric and a skew-symmetric matrix is neither symmetric nor skew-symmetric in general. These properties are useful in tensor algebra and multilinear algebra.

Q: What is the connection between symmetric matrices and the method of least squares?

In the method of least squares, the normal equations involve the matrix ATA, which is always symmetric. The properties of symmetric matrices, such as positive definiteness (if A has full column rank), ensure that the least squares problem has a unique solution and can be solved efficiently.

Q: How can you use the properties of skew-symmetric matrices in computer graphics and robotics?

In 3D computer graphics and robotics, 3x3 skew-symmetric matrices are used to represent cross products and rotations. This representation allows for efficient computation of rotations and is particularly useful in implementing algorithms for 3D transformations and kinematics.

Q: What is the significance of the Schur decomposition for symmetric matrices?

For a symmetric matrix, the Schur decomposition simplifies to the spectral decomposition. This means that not only can a symmetric matrix be upper triangularized by an orthogonal similarity transformation, but it can be fully diagonalized. This property is useful in numerical linear algebra and theoretical analysis.

Q: How do symmetric and skew-symmetric matrices behave under matrix powers?

For a symmetric matrix A, all odd powers (A^(2k+1)) are symmetric, and all even powers (A^(2k)) are symmetric and positive semidefinite. For a skew-symmetric matrix S, all odd powers are skew-symmetric, and all even powers are symmetric and negative semidefinite (except S^0 = I).

Q: How do symmetric and skew-symmetric matrices relate to conservation laws in physics?

In physics, symmetric matrices often represent conservative forces or potentials, while skew-symmetric matrices can represent rotations or angular momenta. This connection between matrix properties and physical conservation laws is crucial in formulating and solving problems in classical and quantum mechanics.

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