Question

IF A is a non- - singular matrix, then show that adj (adj A) = mod of A to the power (n-2 ) A​

Answer 1

Answer:

Step-by-step explanation:

Correction: |A| is known as the determinant of A, not mod of A (when A is a matrix)

We have |A| ≠ 0 ( ∵ A is a non-singular matrix )

We know that,

A(adjA) = |A|.Iₙ ( where I is identity matrix and order of Iₙ = order of A )

Replace A by adjA then

(adjA)(adj(adjA)) = |adjA|.Iₙ = |A|ⁿ⁻¹.Iₙ            (∵ |adjA| = |A|ⁿ⁻¹,if order of A is n )

                                           = Iₙ.|A|ⁿ⁻¹

Pre-multiplying both sides by matrix A,then

A(adjA)(adj(adjA)) = A.Iₙ.|A|ⁿ⁻¹

or,|A|.Iₙ.(adj(adjA)) =A|A|ⁿ⁻¹            ( ∵ A(adjA) = |A|.Iₙ )

or, (adj(adjA)) = |A|ⁿ⁻²A

EXTRA:

A(adjA) = |A|.Iₙ

or, |A(adjA)| = ||A|.Iₙ|

or, |A||adjA| = |A|ⁿ.|Iₙ|          (∵ |kA|=kⁿ|A|,$k \in \mathbb{R}$)

or, |adjA| = |A|ⁿ⁻¹