Matrices & Determinants
Properties of determinants
Grade 12
Question:
<p>If \(\det(A)\)=5, then det(A\)^{-1}\(A\)^TA^{-1}) equals:</p>
<p>(A) 1/5</p>
<p>(B) 1</p>
<p>(C) 5</p>
<p>(D) 1/25</p>
Step-by-Step Solution
Key Concept: Simplify: $A$^{-1}$A$^TA^{-1} = $A$^TA⁻^2 \Rightarrow determinant becomes $\det(A)$ \times (det $A)⁻^2.
1. det(A$^T)=5
2. det(A$⁻^2)=1/25
3. Multiply \to 5 \times (1/25) = 1/5
Correct Answer: 1