Question

Suppose A is a 3 x 3 matrix such that det (A) = 9. Prove det (3 (A-!') is equal to 3

Answer 1

Answer: The proof is done below.

Step-by-step explanation: Given that A is a 3 x 3 matrix such that det (A) = 9.

We are to prove the following :

`det(3A^(-1))=3.`

For a non-singular matrix B of order n, we have two two properties of its determinant :

`(i)~det(B^(-1))=(1)/(det(B)),\\\\\\(ii)~det(kB)=k^ndet(B),~\textup{k is a scalar.}`

Therefore, we get

`det(A^(-1))=(1)/(det(A))=(1)/(9),`

and so,

`det(3A^(-1))~~~~~~~[\textup{since A is of order 3}]\\\\=3^3det(A^(-1))\\\\=27*(1)/(9)\\\\=3.`

Hence proved.