Chapter 3 – Matrices
Overview
This page provides comprehensive Class 12 Maths. Chapter 3 – Matrices - Free study material for Class 12 Maths. NCERT Solutions, Notes, and PYQs.
Symmetric & Skew-Symmetric Matrices (Previous Year Questions)
Class 12 Mathematics | CBSE Previous Year Questions
Q1
2023
00:00
Find the value of $x$ if the matrix $A = \begin{bmatrix}2 & x \\ 3 & 5\end{bmatrix}$ is symmetric.
For a symmetric matrix, $A = A^T$.
So element at (1,2) = element at (2,1).
Hence, $x = 3$.
Final Answer: $x = 3$
Q2
2021
00:00
Find the value of $a$ if $A = \begin{bmatrix}0 & a \\ -a & 0\end{bmatrix}$ is a skew-symmetric matrix.
For a skew-symmetric matrix, $A^T = -A$.
Diagonal elements must be zero and off-diagonal elements must be negatives of each other.
Given matrix already satisfies this for any value of $a$.
Final Answer: $a$ can be any real number.
Q3
2022
00:00
Write whether the matrix $A = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$ is symmetric or skew-symmetric.
$A^T = A$.
Hence, the matrix is symmetric.
Final Answer: (a) Symmetric
Q4
2020
00:00
If $A = \begin{bmatrix}0 & 2 \\ -2 & 0\end{bmatrix}$, then $A$ is a:
Transpose of $A$ is $A^T = \begin{bmatrix}0 & -2 \\ 2 & 0\end{bmatrix}$.
We have $A^T = -A$.
Hence, $A$ is skew-symmetric.
Final Answer: (b) Skew-symmetric matrix
Q5
2024
00:00
If $A = \begin{bmatrix}a & b \\ b & c\end{bmatrix}$, show that $A$ is a symmetric matrix.
Find the transpose of $A$.
$A^T = \begin{bmatrix}a & b \\ b & c\end{bmatrix}$.
Since $A^T = A$, the matrix is symmetric.
Final Answer: $A$ is a symmetric matrix.
Q6
2023
00:00
If $A = \begin{bmatrix}0 & x \\ -x & 0\end{bmatrix}$ and $A^2 = -I$, find the value of $x$.
Compute $A^2$.
$A^2 = \begin{bmatrix}-x^2 & 0 \\ 0 & -x^2\end{bmatrix}$.
Given $A^2 = -I = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}$.
So $x^2 = 1$.
Hence, $x = \pm 1$.
Final Answer: $x = \pm 1$
Q7
2019
00:00
Express the matrix $A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
Use the formula: $A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$.
Here $A^T = \begin{bmatrix}2 & 4 \\ 3 & 5\end{bmatrix}$.
So symmetric part = $\frac{1}{2}(A + A^T)$ and skew-symmetric part = $\frac{1}{2}(A - A^T)$.
Compute both matrices.
Final Answer: $A = S + K$, where $S = \frac{1}{2}(A + A^T)$ and $K = \frac{1}{2}(A - A^T)$.
Q8
2022
00:00
If $A = \begin{bmatrix}a & 2 \\ -2 & b\end{bmatrix}$ is skew-symmetric, find $a$ and $b$.
For a skew-symmetric matrix, diagonal elements are zero.
So $a = 0$ and $b = 0$.
Final Answer: $a = 0, b = 0$
Q9
2023
00:00
Find the symmetric and skew-symmetric parts of the matrix $A = \begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix}$.
Find transpose: $A^T = \begin{bmatrix}1 & 2 \\ 4 & 3\end{bmatrix}$.
Symmetric part $S = \frac{1}{2}(A + A^T)$.
Skew-symmetric part $K = \frac{1}{2}(A - A^T)$.
Compute both matrices.
Final Answer: $S = \frac{1}{2}(A + A^T),\; K = \frac{1}{2}(A - A^T)$
Q10
2021
00:00
Show that the matrix $A = \begin{bmatrix}0 & 1 & 2 \\ -1 & 0 & 3 \\ -2 & -3 & 0\end{bmatrix}$ is skew-symmetric.
Find the transpose $A^T$.
Verify that $A^T = -A$.
Hence, $A$ is skew-symmetric.
Final Answer: $A^T = -A$, so $A$ is skew-symmetric.
Q11
2024
00:00
If $A = \begin{bmatrix}x & 1 \\ -1 & 2\end{bmatrix}$ is neither symmetric nor skew-symmetric, find $x$.
For symmetric: $x$ must satisfy $1 = -1$ (not possible).
For skew-symmetric: diagonal elements must be zero, so $x = 0$ and $2 = 0$ (not possible).
Hence, for all real $x$, the matrix is neither symmetric nor skew-symmetric.
Final Answer: For all real $x$, the matrix is neither symmetric nor skew-symmetric.
Q12
2020
00:00
Find $A + A^T$ and $A - A^T$ for $A = \begin{bmatrix}3 & 5 \\ 1 & 4\end{bmatrix}$.
Find $A^T = \begin{bmatrix}3 & 1 \\ 5 & 4\end{bmatrix}$.
Compute $A + A^T$ and $A - A^T$ by adding and subtracting corresponding elements.
Final Answer: $A + A^T$ and $A - A^T$ obtained by direct computation.
Q13
2019
00:00
Prove that any square matrix $A$ can be expressed as the sum of a symmetric and a skew-symmetric matrix.
Use the identity $A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$.
The first term is symmetric and the second term is skew-symmetric.
Final Answer: $A = S + K$, where $S = \frac{1}{2}(A + A^T)$ and $K = \frac{1}{2}(A - A^T)$.
Q14
2023
00:00
If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix, show that $A + B$ is neither symmetric nor skew-symmetric in general.
Take transpose: $(A + B)^T = A^T + B^T = A - B$.
Since $A + B \ne A - B$, it is neither symmetric nor skew-symmetric.
Final Answer: $A + B$ is neither symmetric nor skew-symmetric in general.
Q15
2022
00:00
If $A = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$, then the matrix is:
Transpose of $A$ is $A^T = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$.
Since $A^T = A$, the matrix is symmetric.
Final Answer: (a) Symmetric matrix
Q16
2024
00:00
If $A = \begin{bmatrix}x & 1 \\ -1 & 2\end{bmatrix}$ is neither symmetric nor skew-symmetric, find $x$.
For symmetric matrix: element (1,2) = element (2,1), so $1 = -1$ which is impossible.
For skew-symmetric matrix: diagonal elements must be zero, so $x = 0$ and $2 = 0$, which is not possible.
Hence, for all real values of $x$, the matrix is neither symmetric nor skew-symmetric.
Final Answer: For all real values of $x$, the matrix is neither symmetric nor skew-symmetric.
Q17
2024
00:00
If $A = \begin{bmatrix}a & b \\ b & c\end{bmatrix}$, show that $A$ is a symmetric matrix.
Transpose of $A$ is $A^T = \begin{bmatrix}a & b \\ b & c\end{bmatrix}$.
Since $A^T = A$, the matrix is symmetric.
Final Answer: $A$ is a symmetric matrix.