To determine the inverse of the given matrix
[tex]\[
\left[\begin{array}{cc}
1 & 3 \\
-1 & 2
\end{array}\right],
\][/tex]
we need to compare it with the provided expressions and identify which one matches the inverse of the matrix.
Let's examine the given options one by one. First, let’s verify the correct inverse by algebraically finding the inverse of the matrix.
The formula for the inverse of a 2x2 matrix [tex]\( \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \)[/tex] is given by:
[tex]\[ \frac{1}{ad - bc} \left[ \begin{array}{cc} d & -b \\ -c & a \end{array} \right] \][/tex]
For our matrix:
[tex]\[ a = 1, b = 3, c = -1, d = 2 \][/tex]
First, calculate the determinant [tex]\( ad - bc \)[/tex]:
[tex]\[ \text{Determinant} = (1)(2) - (3)(-1) = 2 + 3 = 5 \][/tex]
Now, apply the formula for the inverse:
[tex]\[ \frac{1}{5} \left[ \begin{array}{cc} 2 & -3 \\ 1 & 1 \end{array} \right] \][/tex]
Now let's compare this result with the given options:
1. [tex]\(\frac{1}{5}\left[\begin{array}{cc}-2 & -3 \\ 1 & -1\end{array}\right]\)[/tex]:
[tex]\[
\frac{1}{5}\left[\begin{array}{cc}-2 & -3 \\ 1 & -1\end{array}\right] = \left[\begin{array}{cc} -0.4 & -0.6 \\ 0.2 & -0.2 \end{array}\right]
\][/tex]
This does not match our result.
2. [tex]\(\frac{1}{-1}\left[\begin{array}{cc}-2 & -3 \\ 1 & -1\end{array}\right]\)[/tex]:
[tex]\[
\frac{1}{-1}\left[\begin{array}{cc}-2 & -3 \\ 1 & -1\end{array}\right] = \left[\begin{array}{cc} 2 & 3 \\ -1 & 1 \end{array}\right]
\][/tex]
This does not match our result.
3. [tex]\(\frac{1}{-1}\left[\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right]\)[/tex]:
[tex]\[
\frac{1}{-1}\left[\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right] = \left[\begin{array}{cc} -2 & 3 \\ -1 & -1 \end{array}\right]
\][/tex]
This does not match our result.
4. [tex]\(\frac{1}{5}\left[\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right]\)[/tex]:
[tex]\[
\frac{1}{5}\left[\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right] = \left[\begin{array}{cc} 0.4 & -0.6 \\ 0.2 & 0.2 \end{array}\right]
\][/tex]
This matches exactly with our calculated inverse.
Therefore, the expression that represents the inverse of the given matrix is:
[tex]\[
\frac{1}{5}\left[\begin{array}{cc}2 & -3 \\ 1 & 1\end{array}\right]
\][/tex]
And the correct option is:
[tex]\[
\boxed{4}
\][/tex]
