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Q8(i):
For the matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$, verify that (i) $(A + A')$ is a symmetric matrix
Solution :
Step 1: Given Matrix and Transpose Definition
Given the matrix $A$ of order $2 \times 2$:
$A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$
The transpose of a matrix, denoted as $A'$, is obtained by interchanging its rows and columns [Per the definition of Matrix Transposition, where $A'_{ij} = A_{ji}$].
$A' = \begin{bmatrix} 1 & 6 \\ 5 & 7 \end{bmatrix}$
Step 2: Calculating the Sum $(A + A')$
To compute the sum, we perform element-wise addition [Per the Matrix Addition Property]:
$(A + A') = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix} + \begin{bmatrix} 1 & 6 \\ 5 & 7 \end{bmatrix}$
$(A + A') = \begin{bmatrix} (1 + 1) & (5 + 6) \\ (6 + 5) & (7 + 7) \end{bmatrix}$
$(A + A') = \begin{bmatrix} 2 & 11 \\ 11 & 14 \end{bmatrix}$
Step 3: Verification of Symmetry
A matrix $M$ is defined as symmetric if and only if $M' = M$ [By the definition of a Symmetric Matrix]. Let $S = (A + A')$. We must verify if $S' = S$.
Taking the transpose of $S$:
$S = \begin{bmatrix} 2 & 11 \\ 11 & 14 \end{bmatrix}$
$S' = \begin{bmatrix} 2 & 11 \\ 11 & 14 \end{bmatrix}$
Comparing $S'$ and $S$, we observe that $S'_{ij} = S_{ij}$ for all $i, j$ [Verification via identity of matrix components].
Visual Representation of Symmetric Matrix $(A + A')$
Final Solution: Since $(A + A')' = (A + A')$, the matrix $(A + A')$ is proven to be a symmetric matrix.
More Questions from Class 12 Mathematics Matrices EXERCISE 3.3
- Q1(i): Find the transpose of each of the following matrices: (i) $\begin{bmatrix} 5 \\ \frac{1}{2} \\ -1 \end{bmatrix}$
- 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(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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