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Q1(i):
Find the transpose of each of the following matrices: (i) $\begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}$
Solution :
Step 1: Theoretical Foundation of Matrix Transposition
The transpose of a matrix, denoted as $A^T$ or $A'$, is an operation that converts rows into columns and columns into rows. For any matrix $A$ of order $m \times n$, its transpose $A^T$ will have the order $n \times m$. Formally, if $A = [a_{ij}]$, then $A^T = [a_{ji}]$, where the element at the $i$-th row and $j$-th column is shifted to the $j$-th row and $i$-th column [By the definition of matrix transpose].
Step 2: Analysis of the Given Matrix
Let the given matrix be $A$. $A = \begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}$
- Matrix Dimensions: The matrix $A$ consists of 3 rows and 1 column. Therefore, the order is $3 \times 1$.
- Expected Dimensions of $A^T$: Since the order is $m \times n$, the transpose $A^T$ must have an order of $n \times m$, which is $1 \times 3$.
Step 3: Execution of the Transposition
To compute $A^T$, we take the column vector and rewrite it as a row vector:
$A^T = \begin{bmatrix} 5 & \frac{1}{2} & -1 \end{bmatrix}$
Step 4: Geometric/Structural Representation
Final Solution: The transpose of the matrix $\begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}$ is $\begin{bmatrix} 5 & \frac{1}{2} & -1 \end{bmatrix}$.
More Questions from Class 12 Mathematics Matrices EXERCISE 3.3
- Q1(ii): Find the transpose of each of the following matrices: (ii) $\begin{bmatrix} 1 & -1 \\ 2 & 3 \end{bmatrix}$
- Q1(iii): Find the transpose of each of the following matrices: (iii) $\begin{bmatrix} -1 & 5 & 6 \\ \sqrt{3} & 5 & 6 \\ 2 & 3 & -1 \end{bmatrix}$
- Q2(i): If $A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}$, then verify that (i) $(A + B)' = A' + B'$
- Q2(ii): If $A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}$, then verify that (ii) $(A – B)' = A' – B'$
- Q3(i): If $A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, then verify that (i) $(A + B)' = A' + B'$
- Q3(ii): If $A' = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, then verify that (ii) $(A – B)' = A' – B'$
- Q4: If $A' = \begin{bmatrix} -2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 1 & 2 \end{bmatrix}$, then find $(A + 2B)'$
- Q5(i): For the matrices A and B, verify that $(AB)' = B'A'$, where (i) $A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix}$, $B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}$
- Q5(ii): For the matrices A and B, verify that $(AB)' = B'A'$, where (ii) $A = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 5 & 7 \end{bmatrix}$
- Q6(i): If (i) $A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}$, then verify that $A' A = I$
- Q6(ii): If (ii) $A = \begin{bmatrix} \sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha \end{bmatrix}$, then verify that $A' A = I$
- Q7(i): (i) Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.
- Q7(ii): (ii) Show that the matrix $A = \begin{bmatrix} 0 & 1 & -1 \\ -1 & 0 & 1 \\ 1 & -1 & 0 \end{bmatrix}$ is a skew symmetric matrix.
- Q8(i): For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, verify that (i) $(A + A')$ is a symmetric matrix
- Q8(ii): For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, verify that (ii) $(A – A')$ is a skew symmetric matrix
- Q9: Find $\frac{1}{2}(A + A')$ and $\frac{1}{2}(A - A')$, when $A = \begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}$
- Q10(i): Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) $\begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$
- Q10(ii): Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (ii) $\begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix}$
- Q10(iii): Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (iii) $\begin{bmatrix} 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \end{bmatrix}$
- Q10(iv): Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (iv) $\begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}$
- Q11: If $A, B$ are symmetric matrices of same order, then $AB – BA$ is a
- Q12: If $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$, and $A + A' = I$, then the value of $\alpha$ is
CBSE Solutions for Class 12 Mathematics Matrices
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