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