Symmetric Matrix & Skew Symmetric Matrix

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.

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

1. A²ⁿ is a symmetric matrix for n ∈ N.
2. A²ⁿ⁺¹ is a skew-symmetric matrix for n ∈ N
3. kA is also a skew-symmetric matrix, where k ∈ R
4. 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:

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

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

1. A ± B, AB - BA are skew-symmetric matrices.
2. 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.

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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