If A = 1/9 [(-8,1,4)(4,4,7)(1,-8,4)] show that A^-1= A’.

Question

If A = \(\frac{1}{9}\begin{bmatrix} -8&1& 4 \\[0.3em] 4 & 4 & 7\\[0.3em] 1 & -8 & 4 \end{bmatrix}\)show that A^-1= A’.

A = 1/9 [(-8,1,4)(4,4,7)(1,-8,4)]

Answer 1

We have,

To show: A ^-1 = A’

Firstly, we find the Transpose of A, i.e. A.’

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

Calculating |A|

Expanding |A| along C₁, we get

= -1

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

= A’ [from eq. (i)]

Thus, A ^-1 = A’

Hence Proved