Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
If $|A| = -15 + 14 = -1$, find $|A^{adj(A)} - |A^{20}|(A-3I)|$.
Step-by-Step Solution
Key Concept: The determinant of the adjugate matrix satisfies $|adj(A)| = |A|^{n-1}$
Given $|A| = -1$, we have $A^{adj(A)} = |A|^{n-1}(A-3I)$. Computing $|A^{20}| = (-1)^{20} = 1$. Therefore $|A^{adj(A)} - |A^{20}|(A-3I)| = |-1|(1 \cdot (14)) = -1(14) = -14$. The calculation uses the property that $|adj(A)| = |A|^{n-1}$ for an $n \times n$ matrix.
Correct Answer: -14