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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) $(A + B)' = A' + B'$
Solution :
Step 1: Calculating the Sum $A + B$
Given matrices:
$A = \begin{bmatrix} -1 & 2 & 3 \\ 5 & 7 & 9 \\ -2 & 1 & 1 \end{bmatrix}$, $B = \begin{bmatrix} -4 & 1 & -5 \\ 1 & 2 & 0 \\ 1 & 3 & 1 \end{bmatrix}$
To compute the sum $A + B$, we add corresponding elements [Per the definition of Matrix Addition]:
$A + B = \begin{bmatrix} (-1 - 4) & (2 + 1) & (3 - 5) \\ (5 + 1) & (7 + 2) & (9 + 0) \\ (-2 + 1) & (1 + 3) & (1 + 1) \end{bmatrix} = \begin{bmatrix} -5 & 3 & -2 \\ 6 & 9 & 9 \\ -1 & 4 & 2 \end{bmatrix}$Step 2: Finding the Transpose of $(A + B)$
The transpose of a matrix is obtained by interchanging its rows and columns [Per the property $(A^T)_{ij} = A_{ji}$].
$(A + B)' = \begin{bmatrix} -5 & 6 & -1 \\ 3 & 9 & 4 \\ -2 & 9 & 2 \end{bmatrix}$Step 3: Calculating Transposes $A'$ and $B'$
Transposing matrix $A$:
$A' = \begin{bmatrix} -1 & 5 & -2 \\ 2 & 7 & 1 \\ 3 & 9 & 1 \end{bmatrix}$Transposing matrix $B$:
$B' = \begin{bmatrix} -4 & 1 & 1 \\ 1 & 2 & 3 \\ -5 & 0 & 1 \end{bmatrix}$Step 4: Adding $A'$ and $B'$
$A' + B' = \begin{bmatrix} (-1 - 4) & (5 + 1) & (-2 + 1) \\ (2 + 1) & (7 + 2) & (1 + 3) \\ (3 - 5) & (9 + 0) & (1 + 1) \end{bmatrix} = \begin{bmatrix} -5 & 6 & -1 \\ 3 & 9 & 4 \\ -2 & 9 & 2 \end{bmatrix}$Step 5: Comparative Analysis and Verification
| Left Hand Side $(A+B)'$ | Right Hand Side $A' + B'$ |
|---|---|
| $\begin{bmatrix} -5 & 6 & -1 \\ 3 & 9 & 4 \\ -2 & 9 & 2 \end{bmatrix}$ | $\begin{bmatrix} -5 & 6 & -1 \\ 3 & 9 & 4 \\ -2 & 9 & 2 \end{bmatrix}$ |
Since the resulting matrix from the Left Hand Side is identical to the matrix from the Right Hand Side, the property $(A + B)' = A' + B'$ is verified.
Final Solution: The identity $(A + B)' = A' + B'$ is verified as both sides yield $\begin{bmatrix} -5 & 6 & -1 \\ 3 & 9 & 4 \\ -2 & 9 & 2 \end{bmatrix}$.
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(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
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