Question

question 1-3 look at the matrix equation. \begin{bmatrix} 6&4\\ -3&5 end{bmatrix} \times \begin{bmatrix} -9&7\\ 2&8 end{bmatrix} =? which matrix can replace the question mark to make the equation true? \begin{bmatrix} -54&28\\ -6&40 end{bmatrix} \begin{bmatrix} -46&74\\ 37&19 end{bmatrix} \begin{bmatrix} -26&85\\ 35&24 end{bmatrix} \begin{bmatrix} 46&-74\\ 73&21 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=

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

$

Step2: Calculate the first - row elements

For the first row of the product matrix:
The first element is $6\times(-9)+4\times2=-54 + 8=-46$.
The second element is $6\times7+4\times8=42 + 32 = 74$.

Step3: Calculate the second - row elements

For the second row of the product matrix:
The first element is $-3\times(-9)+5\times2 = 27+10=37$.
The second element is $-3\times7+5\times8=-21 + 40 = 19$.

Answer:

$$\begin{bmatrix}-46&74\\37&19\end{bmatrix}$$