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.
Transpose of Matrix (Previous Year Questions)
Class 12 Mathematics | CBSE Previous Year Questions
Q1
2023
00:00
Find the transpose of the matrix $A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$.
Transpose of a matrix is obtained by interchanging rows and columns.
So $A^T = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}$.
Final Answer: $A^T = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}$
Q2
2022
00:00
If $A = \begin{bmatrix}2 & 0 & 1 \\ 3 & -1 & 4\end{bmatrix}$, find $A^T$.
Interchange rows and columns.
So $A^T = \begin{bmatrix}2 & 3 \\ 0 & -1 \\ 1 & 4\end{bmatrix}$.
Final Answer: $A^T = \begin{bmatrix}2 & 3 \\ 0 & -1 \\ 1 & 4\end{bmatrix}$
Q3
2021
00:00
If $A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$, verify that $(A^T)^T = A$.
First find $A^T = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}$.
Now find $(A^T)^T = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$.
Hence, $(A^T)^T = A$.
Final Answer: $(A^T)^T = A$
Q4
2020
00:00
Find $x$ if $A = \begin{bmatrix}1 & x \\ 2 & 3\end{bmatrix}$ and $A^T = A$.
Since $A^T = A$, the matrix is symmetric.
So element (1,2) = element (2,1).
Hence, $x = 2$.
Final Answer: $x = 2$
Q5
2024
00:00
Find the transpose of the zero matrix of order $3 \times 2$.
Transpose of a zero matrix is also a zero matrix.
Order changes from $3 \times 2$ to $2 \times 3$.
Final Answer: A zero matrix of order $2 \times 3$
Q6
2023
00:00
If $A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$, find $A^T$.
Interchange rows and columns.
So $A^T = \begin{bmatrix}a & c \\ b & d\end{bmatrix}$.
Final Answer: $A^T = \begin{bmatrix}a & c \\ b & d\end{bmatrix}$
Q7
2019
00:00
If $A$ is a square matrix, show that $(A + A^T)^T = A + A^T$.
Using property: $(A + B)^T = A^T + B^T$.
So $(A + A^T)^T = A^T + (A^T)^T = A^T + A = A + A^T$.
Final Answer: $(A + A^T)^T = A + A^T$
Q8
2022
00:00
If $A$ is a skew-symmetric matrix, prove that $A^T = -A$.
By definition of skew-symmetric matrix, $A^T = -A$.
Hence proved.
Final Answer: $A^T = -A$
Q9
2024
00:00
Find the transpose of the zero matrix of order $3 \times 2$.
Transpose of a matrix is obtained by interchanging its rows and columns.
Transpose of a zero matrix is also a zero matrix.
Order changes from $3 \times 2$ to $2 \times 3$.
Final Answer: The transpose is a zero matrix of order $2 \times 3$.
Q10
2019
00:00
If $A$ is a square matrix, show that $(A + A^T)^T = A + A^T$.
Using the property: $(M + N)^T = M^T + N^T$.
So, $(A + A^T)^T = A^T + (A^T)^T$.
But $(A^T)^T = A$.
Hence, $(A + A^T)^T = A^T + A = A + A^T$.
Final Answer: $(A + A^T)^T = A + A^T$.
Q11
2022
00:00
If $A$ is a skew-symmetric matrix, prove that $A^T = -A$.
By definition of a skew-symmetric matrix, $A^T = -A$.
Hence proved.
Final Answer: $A^T = -A$.