Sure! Let's evaluate the given matrix expression step by step:
### Step 1: Write down the matrices
We start with two matrices:
[tex]\[
A = \begin{bmatrix} 13 & -12 \\ -22 & 18 \end{bmatrix}
\][/tex]
[tex]\[
B = \begin{bmatrix} -9 & 24 \\ 14 & 8 \end{bmatrix}
\][/tex]
### Step 2: Calculate the difference between the two matrices
To find [tex]\( A - B \)[/tex], subtract the corresponding elements of matrix [tex]\( B \)[/tex] from matrix [tex]\( A \)[/tex]:
[tex]\[
A - B = \begin{bmatrix} 13 & -12 \\ -22 & 18 \end{bmatrix} - \begin{bmatrix} -9 & 24 \\ 14 & 8 \end{bmatrix}
\][/tex]
Performing the element-wise subtraction:
[tex]\[
A - B = \begin{bmatrix} 13 - (-9) & -12 - 24 \\ -22 - 14 & 18 - 8 \end{bmatrix}
\][/tex]
[tex]\[
A - B = \begin{bmatrix} 13 + 9 & -12 - 24 \\ -22 - 14 & 18 - 8 \end{bmatrix}
\][/tex]
[tex]\[
A - B = \begin{bmatrix} 22 & -36 \\ -36 & 10 \end{bmatrix}
\][/tex]
### Step 3: Multiply the resulting matrix by 53
Now, we multiply each element of the resulting matrix by 53:
[tex]\[
53 \cdot \begin{bmatrix} 22 & -36 \\ -36 & 10 \end{bmatrix}
\][/tex]
Performing the scalar multiplication:
[tex]\[
53 \cdot \begin{bmatrix} 22 & -36 \\ -36 & 10 \end{bmatrix} = \begin{bmatrix} 53 \cdot 22 & 53 \cdot -36 \\ 53 \cdot -36 & 53 \cdot 10 \end{bmatrix}
\][/tex]
[tex]\[
53 \cdot \begin{bmatrix} 22 & -36 \\ -36 & 10 \end{bmatrix} = \begin{bmatrix} 1166 & -1908 \\ -1908 & 530 \end{bmatrix}
\][/tex]
### Final Result
Hence, the expression evaluates to:
[tex]\[
53 \left( \begin{bmatrix} 13 & -12 \\ -22 & 18 \end{bmatrix} - \begin{bmatrix} -9 & 24 \\ 14 & 8 \end{bmatrix} \right) = \begin{bmatrix} 1166 & -1908 \\ -1908 & 530 \end{bmatrix}
\][/tex]
So, you have:
[tex]\[
\begin{bmatrix}
53 \left(13 - (-9)\right) & 53 \left(-12 - 24\right) \\
53 \left(-22 - 14\right) & 53 \left(18 - 8\right)
\end{bmatrix} = \begin{bmatrix}
1166 & -1908 \\
-1908 & 530
\end{bmatrix}
\][/tex]
