Certainly! Let's go through the step-by-step solution to perform the matrix addition and multiplication with the given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
### Matrix Addition:
First, let's add the corresponding elements of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
1 & 2 & 3 \\
4 & -2 & 1 \\
0 & -2 & 1
\end{pmatrix}
\][/tex]
Matrix [tex]\( B \)[/tex]:
[tex]\[
B = \begin{pmatrix}
1 & 3 & -2 \\
1 & -4 & 2 \\
0 & 1 & 4
\end{pmatrix}
\][/tex]
The sum of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is calculated as follows:
[tex]\[
C_{\text{add}} = A + B = \begin{pmatrix}
1 + 1 & 2 + 3 & 3 + (-2) \\
4 + 1 & -2 + (-4) & 1 + 2 \\
0 + 0 & -2 + 1 & 1 + 4
\end{pmatrix}
= \begin{pmatrix}
2 & 5 & 1 \\
5 & -6 & 3 \\
0 & -1 & 5
\end{pmatrix}
\][/tex]
So, the result of the matrix addition is:
[tex]\[
C_{\text{add}} = \begin{pmatrix}
2 & 5 & 1 \\
5 & -6 & 3 \\
0 & -1 & 5
\end{pmatrix}
\][/tex]
### Matrix Multiplication:
Next, let's perform the multiplication of matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
The product matrix [tex]\( C_{\text{mul}} \)[/tex] is calculated using the dot product of rows of [tex]\( A \)[/tex] with columns of [tex]\( B \)[/tex].
Matrix [tex]\( A \)[/tex]:
[tex]\[
A = \begin{pmatrix}
1 & 2 & 3 \\
4 & -2 & 1 \\
0 & -2 & 1
\end{pmatrix}
\][/tex]
Matrix [tex]\( B \)[/tex]:
[tex]\[
B = \begin{pmatrix}
1 & 3 & -2 \\
1 & -4 & 2 \\
0 & 1 & 4
\end{pmatrix}
\][/tex]
The product [tex]\( C_{\text{mul}} \)[/tex] is:
[tex]\[
C_{\text{mul}} = A \cdot B
\][/tex]
Let's calculate each element of the resulting matrix:
[tex]\[
C_{\text{mul}} (1, 1) = 1 \cdot 1 + 2 \cdot 1 + 3 \cdot 0 = 1 + 2 + 0 = 3
\][/tex]
[tex]\[
C_{\text{mul}} (1, 2) = 1 \cdot 3 + 2 \cdot -4 + 3 \cdot 1 = 3 - 8 + 3 = -2
\][/tex]
[tex]\[
C_{\text{mul}} (1, 3) = 1 \cdot -2 + 2 \cdot 2 + 3 \cdot 4 = -2 + 4 + 12 = 14
\][/tex]
[tex]\[
C_{\text{mul}} (2, 1) = 4 \cdot 1 + (-2) \cdot 1 + 1 \cdot 0 = 4 - 2 + 0 = 2
\][/tex]
[tex]\[
C_{\text{mul}} (2, 2) = 4 \cdot 3 + (-2) \cdot -4 + 1 \cdot 1 = 12 + 8 + 1 = 21
\][/tex]
[tex]\[
C_{\text{mul}} (2, 3) = 4 \cdot -2 + (-2) \cdot 2 + 1 \cdot 4 = -8 - 4 + 4 = -8
\][/tex]
[tex]\[
C_{\text{mul}} (3, 1) = 0 \cdot 1 + (-2) \cdot 1 + 1 \cdot 0 = 0 - 2 + 0 = -2
\][/tex]
[tex]\[
C_{\text{mul}} (3, 2) = 0 \cdot 3 + (-2) \cdot -4 + 1 \cdot 1 = 0 + 8 + 1 = 9
\][/tex]
[tex]\[
C_{\text{mul}} (3, 3) = 0 \cdot -2 + (-2) \cdot 2 + 1 \cdot 4 = 0 - 4 + 4 = 0
\][/tex]
So, the product matrix [tex]\( C_{\text{mul}} \)[/tex] is:
[tex]\[
C_{\text{mul}} = \begin{pmatrix}
3 & -2 & 14 \\
2 & 21 & -8 \\
-2 & 9 & 0
\end{pmatrix}
\][/tex]
### Conclusion:
The result of the matrix addition is:
[tex]\[
\begin{pmatrix}
2 & 5 & 1 \\
5 & -6 & 3 \\
0 & -1 & 5
\end{pmatrix}
\][/tex]
The result of the matrix multiplication is:
[tex]\[
\begin{pmatrix}
3 & -2 & 14 \\
2 & 21 & -8 \\
-2 & 9 & 0
\end{pmatrix}
\][/tex]
