If A = [(3,-3,4)(2,-3,4)(0,-1,1)] , prove that A^-1 = A^3.

Question

If A = \(\begin{bmatrix} 3& -3 & 4 \\[0.3em] 2 & -3 & 4\\[0.3em] 0 &-1 & 1 \end{bmatrix}\), prove that A ^-1 = A³.

A = [(3,-3,4)(2,-3,4)(0,-1,1)]

Answer 1

We have, A = \(\begin{bmatrix} 3 & -3& 4 \\[0.3em] 2 &-3 & 4 \\[0.3em] 0 & -1 & 1 \end{bmatrix}\)

To show: A^-1 = A³

Firstly, we have to find A ^-1 and A^-1 = \(\frac{adj\,A}{|A|}\)

Calculating |A|

Expanding |A| along C₁, we get

= 3(-3 – (-4)) – 2(-3 – (-4)) + 0 

= 3(- 3 + 4) – 2(-3 + 4) 

= 3(1) – 2(1) 

= 3 – 2 

= 1

Now, we have to find adj A and for that we have to find co-factors:

Calculating A³

= A -1 [from eq. (i)]

Thus, A³ = A ^-1

Hence Proved