If A = [(1,-1,1)(2,-1,0)(1,0,0)], show that A^-1 = A^2.

Question

If A = \(\begin{bmatrix} 1 & -1 &1 \\[0.3em] 2 & -1 & 0\\[0.3em] 1 & 0 & 0 \end{bmatrix}\), show that A^-1 = A².

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

Answer 1

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

To show: A ^-1 = A²

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

Calculating |A| 

Expanding |A| along C₁, we get

= 1(0) – 2(0) + 1(0 – (-1))

= 1(1)

= 1

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

Calculating A²

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

Thus, A² = A ^-1

Hence Proved