Question

If \( A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix} \), then \( A^{-1} \) = ?

• A. \( A - 2A^2 \)
• B. \( 2A - A^2 \)
• C. \( 2A^2 + A \)
• D. \( 2A + A^2 \)

Hint:

To find the inverse of a matrix, explore matrix manipulations like multiplying by its powers or using row operations, depending on the context of the problem.

Answer 1

The Correct Option is B

We are given the matrix \( A \) as:
\[ A = \begin{bmatrix} 1 & 0 & 1\\ 0 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix} \] To find \( A^{-1} \), we can use the matrix inverse formula or perform elementary row operations. However, in this case, based on the structure of the matrix, we observe that the solution involves manipulating \( A \) and its powers. Using the properties of matrix operations, the inverse of the matrix is given by the expression \( 2A - A^2 \). Thus, \( A^{-1} = 2A - A^2 \).