Question

The sum of the matrix \(\begin{bmatrix} 2 & -3 \\ 5 & -7 \end{bmatrix}\) and its multiplicative inverse is

• A. \(\begin{bmatrix} 2 & -3 \\ 5 & -7 \end{bmatrix}\)
• B. \(\begin{bmatrix} -7 & 3 \\ -5 & 2 \end{bmatrix}\)
• C. \(\begin{bmatrix} -5 & 0 \\ 0 & -5 \end{bmatrix}\)
• D. \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

Hint:

Find the inverse using the determinant and adjoint, then add it to the original matrix.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
The inverse of a 2x2 matrix is its adjoint divided by its determinant.

Step 2: Determinant
For \(A=\begin{bmatrix}2&-3\\5&-7\end{bmatrix}\), \(|A|=2(-7)-(-3)(5)=-14+15=1\).

Step 3: Inverse
\(A^{-1}=\frac{1}{1}\begin{bmatrix}-7&3\\-5&2\end{bmatrix}=\begin{bmatrix}-7&3\\-5&2\end{bmatrix}\).

Step 4: Add
\(A+A^{-1}=\begin{bmatrix}2-7&-3+3\\5-5&-7+2\end{bmatrix}=\begin{bmatrix}-5&0\\0&-5\end{bmatrix}\).

Final Answer:
The sum is the matrix with −5 on the diagonal and 0 elsewhere. \[ \boxed{\text{(C) }\begin{bmatrix}-5&0\\0&-5\end{bmatrix}} \]