Use matrix inversion to solve the given system of linear equ

Use matrix inversion to solve the given system of linear equations.

-x+2y-z=0

-x-y+2z=0

2x -z=4

Solution

A = -12-1-1-1220-1

B = 004

X = x₁x₂x₃

A · X = B

so

X = A^-1 · B

Find the inverse matrix using matrix of cofactors (also you can calculate the inverse matrix, using the online Inverse matrix calculator (Gaussian elimination))

Find matrix determinant :

det A = 3

Show detailed calculation of the determinant

The determinant of is not zero, therefore the inverse matrix A^-1 exist. To calculate the inverse matrix find additional minors and cofactors of matrix

Find the minor M₁₁ and the cofactor C₁₁. In matrix A cross out row 1 and column 1.

C₁₁ = (-1)¹⁺¹M₁₁ = 1

Find the minor M₁₂ and the cofactor C₁₂. In matrix A cross out row 1 and column 2.

C₁₂ = (-1)¹⁺²M₁₂ = 3

Find the minor M₁₃ and the cofactor C₁₃. In matrix A cross out row 1 and column 3.

C₁₃ = (-1)¹⁺³M₁₃ = 2

Find the minor M₂₁ and the cofactor C₂₁. In matrix A cross out row 2 and column 1.

C₂₁ = (-1)²⁺¹M₂₁ = 2

Find the minor M₂₂ and the cofactor C₂₂. In matrix A cross out row 2 and column 2.

C₂₂ = (-1)²⁺²M₂₂ = 3

Find the minor M₂₃ and the cofactor C₂₃. In matrix A cross out row 2 and column 3.

C₂₃ = (-1)²⁺³M₂₃ = 4

Find the minor M₃₁ and the cofactor C₃₁. In matrix A cross out row 3 and column 1.

C₃₁ = (-1)³⁺¹M₃₁ = 3

Find the minor M₃₂ and the cofactor C₃₂. In matrix A cross out row 3 and column 2.

C₃₂ = (-1)³⁺²M₃₂ = 3

Find the minor M₃₃ and the cofactor C₃₃. In matrix A cross out row 3 and column 3.

C₃₃ = (-1)³⁺³M₃₃ = 3

Write matrix of cofactors:

C = 132234333

Transposed matrix of cofactors:

C^T = 123333243

Find inverse matrix:

A^-1 = C^Tdet A = 1323111123431

Find a solution:

X = A^-1·B = 1323111123431·004 = 13·0 + 23·0 + 1·41·0 + 1·0 + 1·423·0 + 43·0 + 1·4 = 0 + 0 + 40 + 0 + 40 + 0 + 4 = 444

Answer: