Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
Find the adjugate matrix determinant when $|M| = 2 \times$ Area of triangle with vertices $(a,d)$, $(b,e)$, $(c,f)$ with sides 6, 8, 10.
Step-by-Step Solution
Key Concept: For an $n \times n$ matrix, $|adj(M)| = |M|^{n-1}$
The triangle with sides 6, 8, 10 is a right triangle since $6^2 + 8^2 = 10^2$. Area $= \frac{1}{2} \times 6 \times 8 = 24$. Therefore $|M| = 2 \times 24 = 48$. For a $3 \times 3$ matrix, $|adj(M)| = |M|^{3-1} = 48^2 = 2304$. Thus $|N| = \frac{|adj(M)|^2}{|M|^3} = \frac{2304^2}{48^3} = 288$.
Correct Answer: 288