To find [tex]\( f(B) \)[/tex] where [tex]\( f(x) = x^2 - 4x + 3 \)[/tex], we'll plug the matrix [tex]\( B \)[/tex] into the function. Let's go through the steps:
1. Define the Matrix [tex]\( B \)[/tex]:
[tex]\[
B = \begin{bmatrix}
1 & 2 & 0 \\
3 & -4 & 5 \\
0 & -1 & 2
\end{bmatrix}
\][/tex]
2. Calculate [tex]\( B^2 \)[/tex]:
To calculate [tex]\( B^2 \)[/tex], we multiply [tex]\( B \)[/tex] by itself:
[tex]\[
B \cdot B = \begin{bmatrix}
1 & 2 & 0 \\
3 & -4 & 5 \\
0 & -1 & 2
\end{bmatrix}
\cdot
\begin{bmatrix}
1 & 2 & 0 \\
3 & -4 & 5 \\
0 & -1 & 2
\end{bmatrix}
= \begin{bmatrix}
7 & -6 & 10 \\
-9 & 17 & -10 \\
-3 & 2 & -1
\end{bmatrix}
\][/tex]
3. Calculate [tex]\( 4B \)[/tex]:
Multiply [tex]\( B \)[/tex] by 4:
[tex]\[
4B = 4 \times \begin{bmatrix}
1 & 2 & 0 \\
3 & -4 & 5 \\
0 & -1 & 2
\end{bmatrix}
= \begin{bmatrix}
4 & 8 & 0 \\
12 & -16 & 20 \\
0 & -4 & 8
\end{bmatrix}
\][/tex]
4. Calculate [tex]\( 3I \)[/tex]:
Multiply the 3x3 identity matrix [tex]\( I \)[/tex] by 3:
[tex]\[
3I = 3 \times \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
= \begin{bmatrix}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3
\end{bmatrix}
\][/tex]
5. Combine the Results to Find [tex]\( f(B) \)[/tex]:
Using the function [tex]\( f(x) = x^2 - 4x + 3 \)[/tex], we substitute the matrix [tex]\( B \)[/tex]:
[tex]\[
f(B) = B^2 - 4B + 3I
\][/tex]
Substitute the matrices we calculated:
[tex]\[
f(B) = \begin{bmatrix}
7 & -6 & 10 \\
-9 & 17 & -10 \\
-3 & 2 & -1
\end{bmatrix}
- \begin{bmatrix}
4 & 8 & 0 \\
12 & -16 & 20 \\
0 & -4 & 8
\end{bmatrix}
+ \begin{bmatrix}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 3
\end{bmatrix}
\][/tex]
Perform the matrix subtraction and addition:
[tex]\[
f(B) = \begin{bmatrix}
7 - 4 + 3 & -6 - 8 + 0 & 10 - 0 + 0 \\
-9 - 12 + 0 & 17 - (-16) + 3 & -10 - 20 + 0 \\
-3 - 0 + 0 & 2 - (-4) + 0 & -1 - 8 + 3
\end{bmatrix}
= \begin{bmatrix}
6 & -14 & 10 \\
-21 & 36 & -30 \\
-3 & 6 & -6
\end{bmatrix}
\][/tex]
Thus, the final result for [tex]\( f(B) \)[/tex] is:
[tex]\[
f(B) = \begin{bmatrix}
6 & -14 & 10 \\
-21 & 36 & -30 \\
-3 & 6 & -6
\end{bmatrix}
\][/tex]
