Question

If A is invertible matrix of order 3 and |A| = 5, then find |adj A|

Answer 1

We know that A⁻¹ = `1/|"A"|` adj (A)

∴ A⁻¹ |A| = adj (A)

∴ AA⁻¹ |A| = A adj (A)

∴ I |A| = A adj (A)

∴ det (I |A|) = det (A adj (A))

∴ det (I|A|) = det (A) (adj (A))

∴ `|(5, 0, 0),(0, 5, 0),(0, 0, 5)|` = 5 |adj A|

∴ 5³ = 5 |adj A|

∴ |adj A| = 5² = 25