Question:
If \(A\) is a non-singular matrix and \(A^2-A+I = 0\), then \(A^{-1} = \ldots\)
If \(A\) is a non-singular matrix and \(A^2-A+I = 0\), then \(A^{-1} = \ldots\)
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Multiply the given relation by \(A^{-1}\).
Updated On: Oct 1, 2026
- \(A\)
- \(A-I\)
- \(I-A\)
- \(A+I\)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept
A non-singular matrix has an inverse \(A^{-1}\) with \(AA^{-1}=A^{-1}A=I\).
Step 2: Key Formula or Approach
Multiply both sides of \(A^2-A+I=0\) by \(A^{-1}\).
Step 3: Detailed Explanation
\[ A^{-1}A^2-A^{-1}A+A^{-1}I=0 \]
\[ A-I+A^{-1}=0 \]
\[ A^{-1}=I-A \]
Final Answer:
The inverse is \(I-A\), option (C). \[ \boxed{A^{-1}=I-A\ \text{(C)}} \]
A non-singular matrix has an inverse \(A^{-1}\) with \(AA^{-1}=A^{-1}A=I\).
Step 2: Key Formula or Approach
Multiply both sides of \(A^2-A+I=0\) by \(A^{-1}\).
Step 3: Detailed Explanation
\[ A^{-1}A^2-A^{-1}A+A^{-1}I=0 \]
\[ A-I+A^{-1}=0 \]
\[ A^{-1}=I-A \]
Final Answer:
The inverse is \(I-A\), option (C). \[ \boxed{A^{-1}=I-A\ \text{(C)}} \]
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