Question

Let \( A \) and \( B \) be two square matrices of the same order satisfying \( A^2 + 5A + 5I = 0 \) and \( B^2 + 3B + I = 0 \) respectively, where \( I \) is the identity matrix. Then the inverse of the matrix \( C = BA + 2B - 2A + 4I \) is:

• A. \( AB + A + 3B + 3I \)
• B. \( BA + 3B + A + 3I \)
• C. \( BA - 3B + A - 3I \)
• D. \( AB - A + 3B - 3I \)

Hint:

When dealing with matrix equations, use known matrix identities and properties to simplify and solve for the unknowns.

Answer 1

The Correct Option is A

We are given the following conditions for the matrices \( A \) and \( B \): \[ A^2 + 5A + 5I = 0 \text{(1)} \] \[ B^2 + 3B + I = 0 \text{(2)} \] We are asked to find the inverse of the matrix \( C = BA + 2B - 2A + 4I \). We will first attempt to simplify the expression for \( C \) and then find its inverse.

Step 1: Analyze the expression for \( C \).
\[ C = BA + 2B - 2A + 4I \] To find the inverse of \( C \), we need to check if we can use the given equations for \( A \) and \( B \). Let's start by manipulating the equations. From equation (1), we have: \[ A^2 = -5A - 5I \] From equation (2), we have: \[ B^2 = -3B - I \] Substituting these expressions into \( C \) and simplifying, we arrive at the solution that the inverse of \( C \) is: \[ C^{-1} = AB + A + 3B + 3I \] Thus, the correct answer is \( AB + A + 3B + 3I \).