Question

Given the matrices, \(A\) and \(B\) as \[ A=\begin{bmatrix}1&2\\-2&4\end{bmatrix},\quad B=\begin{bmatrix}3&-2\\5&0\end{bmatrix} \] What will be \(AB\)?

• A. \(\begin{bmatrix}13&-214&4\end{bmatrix}\)
• B. \(\begin{bmatrix}13&214&4\end{bmatrix}\)
• C. \(\begin{bmatrix}-13&214&-4\end{bmatrix}\)
• D. \(\begin{bmatrix}13&2-14&4\end{bmatrix}\)

Hint:

In matrix multiplication, multiply row elements of the first matrix with column elements of the second matrix and then add the products.

Answer 1

The Correct Option is A

Concept: Matrix multiplication is done by multiplying rows of the first matrix with columns of the second matrix. \[ AB=\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix} \]

Step 1: Write the given matrices.
\[ A=\begin{bmatrix}1&2\\-2&4\end{bmatrix} \] \[ B=\begin{bmatrix}3&-2\\5&0\end{bmatrix} \]

Step 2: Find first row first column element.
\[ (1)(3)+(2)(5)=3+10=13 \]

Step 3: Find first row second column element.
\[ (1)(-2)+(2)(0)=-2+0=-2 \]

Step 4: Find second row first column element.
\[ (-2)(3)+(4)(5)=-6+20=14 \]

Step 5: Find second row second column element.
\[ (-2)(-2)+(4)(0)=4+0=4 \]

Step 6: Write the final product matrix.
\[ AB=\begin{bmatrix}13&-2\\14&4\end{bmatrix} \] Therefore, the correct answer is option (A).