Matrices & Determinants
Properties of determinants
Grade 12
Question:
<p>If \(\det(A)\)=k, then det(A\)^n(A\)⁻^n + \(I)(A\)⁻^n - \(I)) equals:</p>
<p>(A) k^n - 1</p>
<p>(B) 1 - k^n</p>
<p>(C) cannot be determined</p>
<p>(D) k^n</p>
Step-by-Step Solution
Key Concept: Even though (X+$I)(X-$I)=X^2-$I$, determinant of resulting sum still not directly computable.
1. Expand: (A$⁻^n + $I)(A$⁻^n - $I) = $A$⁻^2^n - $I$
2. Expression = $A$^n(A$⁻^2^n - $I)
3. Contains subtraction \Rightarrow cannot determine determinant
Correct Answer: 3