Answer to Question #200610 in Linear Algebra for Mpopo

Question

Use Cramer’s rule to solve for x, y and z

2x + y − 3z = 0

4x + 5y + z = 4

x + y − 4z = −1 

Answer 1

First we find the coefficient determinant:

D=$\begin{vmatrix} 2 & 1&-3 \\ 4 & 5&1\\ 1&1&-4 \end{vmatrix}$ =-40-12+1+15-2+16=-22 .

Then we form and evaluate D_x by replacing the first column of values with the answer column:

D_x = $\begin{vmatrix} 0& 1&-3 \\ 4 & 5&1\\ -1&1&-4 \end{vmatrix}$ = 0-12-1-15+0+16=-12.

Form and evaluate Dy by replacing the second column of values with the answer column:

Dy = $\begin{vmatrix} 2& 0&-3 \\ 4 & 4&1\\ 1&-1&-4 \end{vmatrix}$ = -32+12+0+12+2+0=-6 .

Form and evaluate Dz by replacing the third column of values with the answer column:

Dz = $\begin{vmatrix} 2& 1&0 \\ 4 & 5&4\\ 1&1&-1 \end{vmatrix}$ = -10+0+4-8+0+4=-10.

Cramer's rule says that x=D_x/D; y=Dy/D; z=Dz/D.

That is:

x=-12/-22=6/11; y=-6/-22=3/11; z=-10/-22=5/11.

The final answer:

(x,y,z)=(6/11, 3/11, 5/11) .