Let A = [ array cc 1 & 2 -5 & 1 array ] and A ⁻¹=x ~A +y I ₂ , (where I ₂ is unit matrix of order 2), then

Step-by-step solution

array ll & |A|= | array cc 1 & 2 -5 & 1 array |=11 0 & A⁻¹= 1 11 [ array cc 1 & -2 5 & 1 array ] array [ . If A= [ array ll a & b c & d array ] and a d-b c 0 , then .A⁻¹= 1 (a d-b c) [ array cc d & -b -c & a array ] . ] aligned & A ⁻¹=x ~A +y I & 1 11 [ array cc 1 & -2 5 & 1 array ]= [ array cc x & 2 x -5 x & x array ]+ [ array ll y & 0 0 & y array ] & [ array cc 1 11 & -2 11 5 11 & 1 11 array ]= [ array cc x+y & 2 x -5 x & x+y array ] aligned By the equality of matrices, array ll & 2 x= -2 11 and x+y= 1 11 & x= -1