Let's work through the problem step-by-step.
1. Matrix Multiplication [tex]\( AB \)[/tex]:
- First, we multiply matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
[tex]\[
A = \begin{pmatrix}
1 & -2 & 3 \\
-1 & 2 & 1
\end{pmatrix}
\][/tex]
[tex]\[
B = \begin{pmatrix}
2 & 3 \\
3 & 1 \\
1 & 2
\end{pmatrix}
\][/tex]
To find [tex]\( AB \)[/tex], we compute each element of the resulting matrix by taking the dot product of the corresponding row of [tex]\( A \)[/tex] with the corresponding column of [tex]\( B \)[/tex].
[tex]\[
AB = \begin{pmatrix}
(1 \cdot 2) + (-2 \cdot 3) + (3 \cdot 1) & (1 \cdot 3) + (-2 \cdot 1) + (3 \cdot 2) \\
(-1 \cdot 2) + (2 \cdot 3) + (1 \cdot 1) & (-1 \cdot 3) + (2 \cdot 1) + (1 \cdot 2)
\end{pmatrix}
\][/tex]
[tex]\[
AB = \begin{pmatrix}
2 - 6 + 3 & 3 - 2 + 6 \\
-2 + 6 + 1 & -3 + 2 + 2
\end{pmatrix}
\][/tex]
[tex]\[
AB = \begin{pmatrix}
-1 & 7 \\
5 & 1
\end{pmatrix}
\][/tex]
2. Matrix Subtraction [tex]\( AB - 3I \)[/tex]:
- Next, we calculate [tex]\( AB - 3I \)[/tex], where [tex]\( I \)[/tex] is the [tex]\( 2 \times 2 \)[/tex] identity matrix:
[tex]\[
I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
\][/tex]
So, [tex]\( 3I \)[/tex] is:
[tex]\[
3I = \begin{pmatrix}
3 & 0 \\
0 & 3
\end{pmatrix}
\][/tex]
Then, subtract [tex]\( 3I \)[/tex] from [tex]\( AB \)[/tex]:
[tex]\[
AB - 3I = \begin{pmatrix}
-1 & 7 \\
5 & 1
\end{pmatrix}
- \begin{pmatrix}
3 & 0 \\
0 & 3
\end{pmatrix}
\][/tex]
[tex]\[
AB - 3I = \begin{pmatrix}
-1 - 3 & 7 - 0 \\
5 - 0 & 1 - 3
\end{pmatrix}
\][/tex]
[tex]\[
AB - 3I = \begin{pmatrix}
-4 & 7 \\
5 & -2
\end{pmatrix}
\][/tex]
3. Finding the Determinant:
- Finally, we find the determinant of the resultant matrix [tex]\( AB - 3I \)[/tex]:
[tex]\[
\text{Det}(AB - 3I) = \text{Det}\begin{pmatrix}
-4 & 7 \\
5 & -2
\end{pmatrix}
\][/tex]
The determinant of a [tex]\( 2 \times 2 \)[/tex] matrix [tex]\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\)[/tex] is given by [tex]\( ad - bc \)[/tex]. Applying this formula:
[tex]\[
\text{Det}\begin{pmatrix}
-4 & 7 \\
5 & -2
\end{pmatrix} = (-4)(-2) - (7)(5)
\][/tex]
[tex]\[
= 8 - 35
\][/tex]
[tex]\[
= -27
\][/tex]
Therefore, the determinant of [tex]\( AB - 3I \)[/tex] is [tex]\(-27\)[/tex].
