# <span lang="EN-US

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 R2R2+R1

Using

Using R1R1-2R2 and R3R3+2R2

Using R35R3

Using and

Here A-1A=I and hence

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