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:
Show Hint
- \( AB + A + 3B + 3I \)
- \( BA + 3B + A + 3I \)
- \( BA - 3B + A - 3I \)
- \( AB - A + 3B - 3I \)
The Correct Option is A
Solution and Explanation
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 \( AB + A + 3B + 3I \).
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