Answer to Question #111229 in Linear Algebra for Mzwandile

Question

Solve the following system of linear equations by using the inverse matrix
method: (10)
x + y + z = 4
-2x - y + 3z = 1
y + 5z = 9

Answer 1

Solve the following system of linear equations using an inverse matrix

$x+y+z=4\\ -2x-y+3z=1\\ y+5z=9$

Solution

$A=\begin{pmatrix} 1 & 1&1 \\ -2 & -1&3\\ 0&1&5 \end{pmatrix}$

$B=\begin{pmatrix} 4 \\ 1 \\ 9 \end{pmatrix}$

$X=\begin{pmatrix} y \\ x\\ z \end{pmatrix}$

$A∙X=B$

that is

X=A^-1 ∙B

Find the determinant of matrix A:

$detA=\begin{vmatrix} 1 & 1&1 \\ -2 & -1&3\\ 0&1&5 \end{vmatrix}= 1·(-1)·5 + 1·3·0 + 1·(-2)·1 - 1·(-1)·0 - 1·3·1 - 1·(-2)·5 =-5 + 0 - 2 - 0 - 3 + 10 = 0$

Answer: Since the determinant of the matrix is zero, the system has no solution.