Matrices & Determinants
Properties of determinants
Grade 12
Question:
<p>If \(\det(A)\)=k, then det(A\)^n(I\) + \(A\)⁻^n)(I\) - \(A\)⁻^n)) 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: Using identity (I$+X)(I$-X)=$I$-X^2 gives $A$^n(I$-$A$⁻^2^n), still a difference.
1. (I$ + $A$⁻^n)(I$ - $A$⁻^n) = $I$ - $A$⁻^2^n
2. Expression = $A$^n(I$ - $A$⁻^2^n)
3. Contains subtraction \Rightarrow cannot determine
Correct Answer: 3