Q25 of 52 Page 1

If , Find A-1. Hence, solve the system of equations x+y+z=6, x+2z=7, 3x+y+z=12.

Or


Find the inverse of the following matrix using elementary operations.


Given that


Then (Expanding along R1)


Calculating the Cofactors of matrix A, we get


C11=-2, C12=5, C13=1


C21=0, C22=-2, C23=2


C31=2, C32=-1, C33=-1


Adjoint of a matrix A is defined as the transpose of Cofactor matrix of matrix A.


Hence,



Inverse of a matrix A is defined as a matrix A-1 such that


AA-1=I=A-1A.


A-1 is also equal to , |A|≠0.


Hence,



Given that


x + y + z=6


x+2z=7


3x+y+z=12


Hence


Pre-multiplying with A-1 both sides, we get



or


Hence x=3, y=1 and z=2.


Or


Given that


We know that A=IA


Hence,


Using R2→R2+R1



Using



Using R1→R1-2R2 and R3→R3+2R2



Using R3→5R3



Using and



Here A-1A=I and hence


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Or


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