Question

If A is an invertible matrix, such that A² - A + I = 0, then the inverse of A is :

• A. A^-2
• B. I - A, where I is an identify matrix of order 2
• C. 0
• D. A

Answer 1

The Correct Option is B

To find the inverse of matrix \( A \), we start with the given equation:

\( A^2 - A + I = 0 \).

This equation can be rearranged to express \( A^2 \):

\( A^2 = A - I \).

Assuming \( A \) is invertible, we can multiply both sides of this equation by \( A^{-1} \):

\( A^{-1}A^2 = A^{-1}(A - I) \).

Since \( A^{-1}A = I \), the left-hand side simplifies to:

\( IA = A \).

The right-hand side simplifies as follows:

\( IA - A^{-1}I = A - A^{-1} \).

So we have:

\( A = (A - I) \).

Rearranging terms gives us:

\( I = A \) or \( A^{-1} = I - A \).

Therefore, the inverse of matrix \( A \) is:

I - A, where \( I \) is the identity matrix of order 2.