Matrices & Determinants Question - None

Question

If A is a square matrix of order 5 and $2A^{-1} = A^T$, then the remainder when $|\text{adj}(\text{adj}(\text{adj}\, A))|$ is divided by 7 is

Accepted Answer

Step 1: Find |A| from the condition 2A⁻¹ = Aᵀ

Taking determinants: |2A⁻¹| = |Aᵀ|

2⁵|A⁻¹| = |A| (since |2A⁻¹| = 2⁵|A⁻¹| and |Aᵀ| = |A|)

32 · (1/|A|) = |A|

|A|² = 32, so |A| = ±4√2

Since we need integer answers mod 7, reconsider: |A|² = 32 implies |A|⁴ = 1024

Step 2: Apply determinant formula for adjugates iteratively

For square matrix M of order n: |adj(M)| = |M|^(n-1)

Let |A| = d. Then:

|adj(A)| = |A|⁴ = d⁴

|adj(adj(A))| = |adj(A)|³ = (d⁴)³ = d¹²

|adj(adj(adj(A)))| = |adj(adj(A))|⁴ = (d¹²)⁴ = d⁴⁸

Step 3: Calculate d⁴⁸ mod 7

From |A|² = 32: |A|⁴ = 1024 ≡ 2 (mod 7)

So d⁴ ≡ 2 (mod 7)

d⁴⁸ = (d⁴)¹² ≡ 2¹² (mod 7)

2¹² = 4096 = 585 × 7 + 1 ≡ 1 (mod 7)

∴ Answer: D (remainder is 1)

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