Using matrices, solve the following system of equations:
2x - 3y + 5z = 11

3x + 2y - 4z = -5

x + y - 2z = -3

Question

Using matrices, solve the following system of equations:2x - 3y + 5z = 113x + 2y - 4z = -5x + y - 2z = -3

Philoid · Accepted Answer

The above system of equations can be expressed as matrix equation

AX = B

∴ X = A^-1B

Now,

|A| = 2[2(-2) – (-4)(1)] + 3[3(-2) – (-4)(1)] + 5[3(1) – 2(1)]

= 2(-4 + 4) + 3(-6 + 4) + 5(1)

= -6 + 5 = -1 ≠ 0

As, determinant ≠ 0, A^-1 exists

_{We know,}

_A^-1 = |A|(adj.A), where

_Minor _{of an element a}_ij of the determinant of matrix A is the determinant obtained by deleting ith row and jth column and denoted by M_ij

_and

_Cofactor _{of a}_ij of given by A_ij = (– 1)^i+j M_ij

_And

If then,

where, A_ij is cofactor of a_ij

Calculating for

We get,

a₁₁ = 2, A₁₁ = 0

a₁₂ = -3, A₁₂ = 2

a₁₃ = 5, A₁₃ = 1

a₂₁ = 3, A₂₁ = -1

a₂₂ = 2, A₂₂ = -9

a₂₃ = -4, A₂₃ = -5

a₃₁ = 1, A₃₁ = 2

a₃₂ = 1, A₃₂ = 23

a₃₃ = -2, A₃₃ = 13

and

∴

X = A^-1B