Question

If \[ A= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \qquad B= \begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix} \] then \(AB\) is equal to:

• A. \[ \begin{bmatrix} 4 & 10 \\ 10 & 20 \end{bmatrix} \]
• B. \[ \begin{bmatrix} 4 & 10 \\ 10 & 15 \end{bmatrix} \]
• C. \[ \begin{bmatrix} 4 & 10 \\ 10 & 20 \end{bmatrix} \]
• D. \[ \begin{bmatrix} 4 & 10 \\ 10 & 15 \end{bmatrix} \]

Hint:

For matrix multiplication: \[ (\text{Row})\times(\text{Column}) \] Multiply corresponding entries and add the products carefully.

Answer 1

The Correct Option is A

Concept: Matrix multiplication is performed row-wise and column-wise. If \[ A=[a_{ij}] \quad \text{and} \quad B=[b_{ij}], \] then each entry of \(AB\) is obtained by multiplying the corresponding row of \(A\) with the corresponding column of \(B\).

Step 1: Write the matrices clearly. \[ A= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] and \[ B= \begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix} \] We compute: \[ AB = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix} \]

Step 2: Find the first row entries. First row, first column: \[ (1)(2)+(2)(1) = 2+2 = 4 \] First row, second column: \[ (1)(0)+(2)(5) = 0+10 = 10 \] Thus first row becomes: \[ [4 \quad 10] \]

Step 3: Find the second row entries. Second row, first column: \[ (3)(2)+(4)(1) = 6+4 = 10 \] Second row, second column: \[ (3)(0)+(4)(5) = 0+20 = 20 \] Thus second row becomes: \[ [10 \quad 20] \]

Step 4: Write the final matrix. Hence, \[ AB= \begin{bmatrix} 4 & 10 \\ 10 & 20 \end{bmatrix} \] Therefore, \[ \boxed{ AB= \begin{bmatrix} 4 & 10 \\ 10 & 20 \end{bmatrix} } \]