Question

1. \\(\

$$\begin{bmatrix}-5&-5\\\\-1&2\\end{bmatrix}$$

\cdot\

$$\begin{bmatrix}-2&-3\\\\3&5\\end{bmatrix}$$

\\)

Explanation:

Step1: Recall matrix multiplication rule

For two matrices \( A =

$$\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$$

\) and \( B =

$$\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$$

\), the product \( AB=

$$\begin{bmatrix} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \end{bmatrix}$$

\)

Let \( A=

$$\begin{bmatrix}-5&-5\\-1&2\end{bmatrix}$$

\) and \( B =

$$\begin{bmatrix}-2&-3\\3&5\end{bmatrix}$$

\)

Step2: Calculate first element (1,1)

\( a_{11}b_{11}+a_{12}b_{21}=(-5)\times(-2)+(-5)\times3 = 10 - 15=-5 \)

Step3: Calculate first element (1,2)

\( a_{11}b_{12}+a_{12}b_{22}=(-5)\times(-3)+(-5)\times5=15 - 25=-10 \)

Step4: Calculate second element (2,1)

\( a_{21}b_{11}+a_{22}b_{21}=(-1)\times(-2)+2\times3 = 2 + 6 = 8 \)

Step5: Calculate second element (2,2)

\( a_{21}b_{12}+a_{22}b_{22}=(-1)\times(-3)+2\times5=3 + 10 = 13 \)

Answer:

\(

$$\begin{bmatrix}-5&-10\\8&13\end{bmatrix}$$

\)