Question

If \(A=\begin{bmatrix} -2&6\\-5&-1\end{bmatrix}\) then \(A^{-1}\) is:

• A. \(\frac{1}{28}\begin{bmatrix} 2&-5\\6&-1\end{bmatrix}\)
• B. \(\frac{1}{32}\begin{bmatrix} -1&6\\-5&-2\end{bmatrix}\)
• C. \(\frac{1}{32}\begin{bmatrix} -1&-6\\5&-2\end{bmatrix}\)
• D. \(\frac{-1}{28}\begin{bmatrix} -1&-6\\-5&-2\end{bmatrix}\)

Answer 1

The Correct Option is C

To find the inverse of matrix \(A=\begin{bmatrix} -2 & 6 \\ -5 & -1 \end{bmatrix}\), we use the formula for the inverse of a 2x2 matrix, given by:
\(A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\) where \(a, b, c, d\) are elements of the matrix and \(\text{det}(A) = ad - bc\).
Let \(a = -2, b = 6, c = -5, d = -1\). First, calculate the determinant:
\(\text{det}(A) = (-2)(-1) - (6)(-5) = 2 + 30 = 32\).
Now determine the inverse matrix:
\(A^{-1} = \frac{1}{32} \begin{bmatrix} -1 & -6 \\ 5 & -2 \end{bmatrix}\).
The correct answer is:
\(\frac{1}{32}\begin{bmatrix} -1 & -6 \\ 5 & -2 \end{bmatrix}\).