Question

For a nonsingular matrix a and nonzero scalar β, show that (βa) −1 = 1 β a

Answer 1

To show that (βa)^(-1) = (1/β)a^(-1), we need to show that their product is equal to the identity matrix I.

First, let's compute the product of (βa) and (1/β)a^(-1):

(βa)(1/β)a^(-1) = (1/β)(βaa^(-1)) = (1/β)(I) = 1/β * I

So we see that (βa)(1/β)a^(-1) = 1/β * I. Now we need to show that the right-hand side is equal to the identity matrix I.

We have:

(1/β) * I = (1/β) * a * a^(-1) = a * (1/β) * a^(-1)

So (1/β) * I and a * (1/β) * a^(-1) are equal. Therefore, we have:

(βa)(1/β)a^(-1) = a * (1/β) * a^(-1) = I

This shows that (βa)^(-1) = (1/β)a^(-1), as required.

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