Find the inverse of \(\begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}\)
Answer Details
To find the inverse of a matrix, we need to use the formula:
\[A^{-1} = \frac{1}{\det(A)}\text{adj}(A)\]
where \(\det(A)\) is the determinant of matrix \(A\) and \(\text{adj}(A)\) is the adjugate of matrix \(A\).
In this case, we have:
\[A = \begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}\]
The determinant of \(A\) is:
\[\det(A) = \begin{vmatrix} 3 & 5 \\ 1 & 2 \end{vmatrix} = (3\times2) - (5\times1) = 1\]
The adjugate of \(A\) is:
\[\text{adj}(A) = \begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix}\]
Therefore, the inverse of \(A\) is:
\[A^{-1} = \frac{1}{\det(A)}\text{adj}(A) = \frac{1}{1}\begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix}\]
So the answer is (B) \(\begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix}\).