Question

If A is a square matrix of order 3 such that |A|=3, then write the value of |adj(adjA)|.​

Answer 1

Given : If A is a square matrix of order 3 such that |A| = 3.

To find : The value of |adj(adj(A))|

solution : let's derive the formula,

A is a square matrix of order n ( n × n).

we know, A adj(A) = |A| I , where I is unity matrix.

|A adj (A) | = | |A| I |

⇒|A| |adj(A)| = |A|ⁿ [ as n is order of matrix ]

⇒|adj(A)| = |A|ⁿ¯¹

now |adj( adj(A))| = |adj(A)|ⁿ¯¹

= ||A|ⁿ¯¹|ⁿ¯¹ = |A|^(n - 1)²

Therefore |adj(adj(A))| = |A|^(n - 1)²

here, |A| = 3 and n = 3

so, |adj(adj(A))| = (3)^(3 - 1)² = 3⁴ = 81

Therefore the value of |adj(adj(A))| is 81.

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