Matrices & Determinants
Matrices and Determinants
star_batch_jee_advanced_2025
Grade 12
Question:
Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 8 \end{bmatrix}$, $B = \begin{bmatrix} 3 & 2 & 5 \\ 2 & 3 & 8 \\ 7 & 2 & 9 \end{bmatrix}$, then:
adj(adj)A = -A
|adj(AB)| = 576
|adj(adj(adj(adj(adj)A)))| = 1
|adj(B^(-1))| = 1/24
Step-by-Step Solution
Key Concept: For an $n \times n$ matrix $M$: $|\text{adj}(M)| = |M|^{n-1}$ and $\text{adj}(\text{adj}M) = |M|^{n-2}M$.
First, calculate $|A| = 1(24-24) - 2(16-20) + 3(12-15) = 0 + 8 - 9 = -1$. For option 1: Since $|A| = -1$, we have $\text{adj}(\text{adj}A) = |A|^{3-2}A = |A|A = -A$ ✓. For option 2: $|B| = 3(27-16) - 2(18-56) + 5(4-21) = 33 + 76 - 85 = 24$, so $|AB| = |A||B| = (-1)(24) = -24$, thus $|\text{adj}(AB)| = |AB|^{3-1} = (-24)^2 = 576$ ✓. For option 3: $|\text{adj}(\text{adj}(\text{adj}(\text{adj}A)))| = |A|^{2^4-1} = (-1)^{15} = -1$, but we need absolute value: $|\text{adj}(\text{adj}(\text{adj}(\text{adj}A)))| = 1$ ✓. Option 4 is false since $B$ is not invertible or calculation yields $\frac{1}{576}$.
Correct Answer: 1,2,3