Question

If \(A+I=\begin{pmatrix} 3 & -2 \\ 1 & -1 \end{pmatrix}\), then the inverse of the matrix \((A-I)\) is:

• A. \(\begin{pmatrix} 1 & -2 \\ 1 & -3 \end{pmatrix}\)
• B. \(\begin{pmatrix} 3 & -2 \\ 1 & -1 \end{pmatrix}\)
• C. \(\begin{pmatrix} -3 & 2 \\ -1 & 1 \end{pmatrix}\)
• D. \(\begin{pmatrix} -3 & 2 \\ -1 & -1 \end{pmatrix}\)

Hint:

Subtract 2I from the given matrix to get A minus I, then use the 2x2 inverse formula.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Obtain \(A-I\) from \(A+I\) and then find its inverse using the adjoint and determinant.

Step 2: Find A minus I
\(A-I=(A+I)-2I=\begin{pmatrix}3&-2\\1&-1\end{pmatrix}-\begin{pmatrix}2&0\\0&2\end{pmatrix}=\begin{pmatrix}1&-2\\1&-3\end{pmatrix}\).

Step 3: Determinant and adjoint
\(|A-I|=(1)(-3)-(-2)(1)=-1\).
\(\operatorname{adj}(A-I)=\begin{pmatrix}-3&2\\-1&1\end{pmatrix}\).

Step 4: Inverse
\((A-I)^{-1}=\frac{1}{-1}\begin{pmatrix}-3&2\\-1&1\end{pmatrix}=\begin{pmatrix}3&-2\\1&-1\end{pmatrix}\), which is the second option.

Final Answer:
The inverse of A − I is the matrix with rows (3, −2) and (1, −1). \[ \boxed{\text{(B) }\begin{pmatrix}3&-2\\1&-1\end{pmatrix}} \]