Show that a matrix A is invertible, if and only if A is non-singular.

Asked by Topperlearning User | 2nd May, 2016, 09:35: AM

Expert Answer:

Let A square matrix of order n, then there exists a square matrix of order n such that AB = I, where I is the identity matrix of order n.

N o w space A B equals I
rightwards double arrow open vertical bar A B close vertical bar equals open vertical bar I close vertical bar
rightwards double arrow open vertical bar A close vertical bar times open vertical bar B close vertical bar equals 1
rightwards double arrow open vertical bar A close vertical bar not equal to 0
rightwards double arrow A space i s space n o n minus sin g u l a r

C o n v e r s e l y comma space A space i s space n o n minus sin g u l a r comma space t h a t space i s space open vertical bar A close vertical bar not equal to 0
N o w comma space A open parentheses a d j space A close parentheses equals open vertical bar A close vertical bar I equals open parentheses a d j space A close parentheses A
rightwards double arrow A open parentheses fraction numerator 1 over denominator open vertical bar A close vertical bar end fraction a d j space A close parentheses equals open parentheses fraction numerator 1 over denominator open vertical bar A close vertical bar end fraction a d j space A close parentheses A equals I
rightwards double arrow A B equals B A equals I comma space w h e r e space B equals fraction numerator 1 over denominator open vertical bar A close vertical bar end fraction a d j space A
rightwards double arrow A space i s space i n v e r t i b l e.

Answered by  | 2nd May, 2016, 11:35: AM