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Matrices & Determinants

Adjugate determinant identity

nta_pyq_2025_apr

Grade 12

Question:

Let A be a matrix of order 3 $\times$ 3 and |A| = 5. If |2 adj(3 A adj(2 A))| = 2$\alpha$$\cdot$3$\beta$$\gamma$$\cdot$5$\alpha$,$\beta$,$\gamma$$\ in $N then$\alpha$+$\beta$+$\gamma$is equal to

$25$

$26$

$27$

$28$

Step-by-Step Solution

Key Concept: Apply the matrix property for adjugate determinant identity and reduce it to determinant or parameter equations.

|2 adj(3 A adj(2 A))| 3 2 (3) 2$\cdot$|3 A adj(2 A)| 2 2 2 3 3 2$\cdot$(3 )$\cdot$| A|$\cdot$| adj(2 A)| 2 3 6 2 2 2$\cdot$3$\cdot$| A|$\cdot$(|2 A| ) 2 2 2 2 3 6 3 2$\cdot$3$\cdot$| A| [(2 )$\cdot$| A| ] 3 6 2 12 4 2$\cdot$3$\cdot$| A|$\cdot$2$\cdot$| A| 15 6 6 2$\cdot$3$\cdot$| A| 15 6 6$\alpha$$\beta$$\gamma$2$\cdot$3$\cdot$$5 = 2$$\cdot$3$\cdot$5$\alpha$= 15,$\beta$= 6,$\gamma$= 6$\alpha$+$\beta$+$\gamma$= 27

Correct Answer: 3