Solve the following system of equations by matrix method:

2y – z = 1

x – y + z = 2

2x – y = 0

The given system can be written in matrix form as:

AX = B

Now, |A| = 0

= 0 + 4 – 1

= 3

So, the above system has a unique solution, given by

X = A ^{– 1}B

Cofactors of A are:

C₁₁ = (– 1)^{1 + 1} 1 – 0 = 1

C₂₁ = (– 1)^{2 + 1}1 – 2 = 1

C₃₁ = (– 1)^{3 + 1}0 + 1 = 1

C₁₂ = (– 1)^{1 + 2} – 2 – 0 = 2

C₂₂ = (– 1)^{2 + 1} – 1 – 0 = – 1

C₃₂ = (– 1)^{3 + 1} 0 – 2 = 2

C₁₃ = (– 1)^{1 + 2} 4 – 0 = 4

C₂₃ = (– 1)^{2 + 1} 2 – 0 = – 2

C₃₃ = (– 1)^{3 + 1} – 1 + 2 = 1

adj A =

=

A ^{– 1} =

Now, X = A ^{– 1}B =

X =

X =

X =

Hence, X = 1,Y = 2 and Z = 3