Matrices & Determinants
Properties of Inverse Matrices
Grade 12
Question:
<p>If <i>A</i> and <i>B</i> are two nonsingular matrices of the same order such that \(B^r = I\), for some positive integer \(r > 1\), then \(A^{-1} B^{r-1} A - A^{-1} B^{-1} A =\)</p>
<p>\(I\)</p>
<p>\(2I\)</p>
<p>\(O\)</p>
<p>\(-I\)</p>
Step-by-Step Solution
Key Concept: Since B^r = I, we have B^(r-1) = B^(-1), which allows direct substitution and simplification using associativity of matrix multiplication and the property that A^(-1)A = I.
<p><strong>Step 1:</strong> From the given condition B^r = I, multiply both sides by B^(-1):</p><p>B^r · B^(-1) = I · B^(-1)</p><p>B^(r-1) = B^(-1)</p><p><strong>Step 2:</strong> Substitute B^(r-1) = B^(-1) into the given expression:</p><p>A^(-1)B^(r-1)A - A^(-1)B^(-1)A = A^(-1)B^(-1)A - A^(-1)B^(-1)A</p><p><strong>Step 3:</strong> Simplifying:</p><p>= 0</p><p><strong>∴ Answer: C (which is 0)</strong></p>
Correct Answer: C