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Let A be a 3x3 matrix with |A| = 2. Match each expression in List-I with its value in List-II. List-I: (P) |3 * adj(-2 * adj(adj(A)))| (Q) |2 * adj(3 * adj(adj(2A)))| (R) |adj(adj(adj(3A)))| (S) |adj(adj(3 * adj(A)))| List-II: (1) 2¹⁴ * 3³ (2) 3¹² * 2⁸ (3) 2³⁵ * 3⁶ (4) 3²⁴ * 2⁸

1. P->1, Q->3, R->4, S->2
2. P->2, Q->1, R->3, S->4
3. P->3, Q->4, R->2, S->1
4. P->4, Q->2, R->1, S->3

Correct answer: P->1, Q->3, R->4, S->2

Solution

(P): Let M2=adj(adj(A)), |M2|=|A|⁴=16. adj(-2*M2)=(-2)²*adj(M2), |adj(-2*M2)|=64*|M2|²=64*256=2¹⁴. Then |3*that_matrix|=3³*2¹⁴=2¹⁴*3³. Matches (1). (Q): |2A|=8*2=2⁴. adj(2A) has det 2⁸. adj(adj(2A)) has det 2¹⁶. adj(3*that)=3²*adj(...), det=3⁶*2³². |2*result|=2³*3⁶*2³²=2³⁵*3⁶. Matches (3). (R): |3A|=3³*2=2*3³. |adj(3A)|=(2*3³)²=2²*3⁶. |adj(adj(3A))|=2⁴*3¹². |adj(adj(adj(3A)))|=2⁸*3²⁴=3²⁴*2⁸. Matches (4). (S): |adj(A)|=4=2². |3*adj(A)|=3³*2²=2²*3³. |adj(3*adj(A))|=(2²*3³)²=2⁴*3⁶. |adj(adj(3*adj(A)))|=2⁸*3¹²=3¹²*2⁸. Matches (2).

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