Question

question
find the product of the following two matrices.
\\(\

$$\begin{bmatrix} 4 & -1 \\\\ -4 & 0 \\end{bmatrix}$$

\

$$\begin{bmatrix} -1 & 1 \\\\ -4 & 0 \\end{bmatrix}$$

\\)
answer attempt 2 out of y

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 \( AB \) is given by:
\[
AB=

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

\]
Here, \( A=

$$\begin{bmatrix}4&-1\\-4&0\end{bmatrix}$$

\) and \( B =

$$\begin{bmatrix}-1&1\\-4&0\end{bmatrix}$$

\)

Step2: Calculate the (1,1) entry

\( a_{11}b_{11}+a_{12}b_{21}=4\times(-1)+(-1)\times(-4)= - 4 + 4=0 \)

Step3: Calculate the (1,2) entry

\( a_{11}b_{12}+a_{12}b_{22}=4\times1+(-1)\times0 = 4+0 = 4 \)

Step4: Calculate the (2,1) entry

\( a_{21}b_{11}+a_{22}b_{21}=(-4)\times(-1)+0\times(-4)=4 + 0=4 \)

Step5: Calculate the (2,2) entry

\( a_{21}b_{12}+a_{22}b_{22}=(-4)\times1+0\times0=-4 + 0=-4 \)

Answer:

\(

$$\begin{bmatrix}0&4\\4&-4\end{bmatrix}$$

\)