If A is 3 × 3 invertible matrix, then show that for any scalar k (non-zero),kA is invertible and (kA)¹ = 1/k.A^-1.

Question

If A is 3 × 3 invertible matrix, then show that for any
scalar k (non-zero),kA is invertible and (kA)1 = 1/k
A-1

Prasanta · Accepted Answer

We know that,

kA-1=1kA×adjkA -----------(1)

⇒kA-1=1kA×kn-1×adj A , where n is the degree of the matrix (here 3)
[since kA=knA]

⇒kA-1=1k3A×k2×adj A

⇒kA-1=1k×1A×adj A

⇒kA-1=1k×A-1

Hence Proved

If A is 3 × 3 invertible matrix, then show that for any scalar k (non-zero),kA is invertible and (kA)1 = 1/k.A-1.