Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7. 2x - y + 4z = -2 3x + 2y - z = -3 x + 4y - 2z = 17

Accepted Answer

Write the system of equations in matrix form as $$A\mathbf{x} = \mathbf{b}$$, where $$A is $$the coefficient matrix, $$\mathbf{x} is $$the column vector of variables, and $$\mathbf{b} is $$the constants vector. For this system, $$A = \begin{bmatrix} 2 & -1 & 4 \ 3 & 2 & -1 \ 1 & 4 & 2 \end{bmatrix}$$ and $$\mathbf{b} = \begin{bmatrix} -2 \ -3 \ 17 \end{bmatrix}. $$Calculate the determinant $$D of $$the coefficient matrix $$A. $$This is done by expanding the determinant of the $$3 	imes 3$$ matrix $$A$$ using cofactor expansion or any preferred method. If $$D 
eq 0$$, proceed to find the determinants $$D_x$$, $$D_y$$, and $$D_z by $$replacing the respective columns of $$A$$ with the vector $$\mathbf{b}. $$Specifically, $$D_x is $$found by replacing the first column of $$A$$ with $$\mathbf{b}$$, $$D_y by $$replacing the second column, and $$D_z by $$replacing the third column. Use Cramer's rule to find the solutions for $$x$$, $$y$$, and $$z by $$computing $$x = \frac{D_x}{D}$$, $$y = \frac{D_y}{D}$$, and $$z = \frac{D_z}{D}. If$$ $$D = 0$$, Cramer's rule cannot be applied. In that case, use another method such as substitution, elimination, or matrix row reduction (Gaussian elimination) to determine whether the system has infinitely many solutions or no solution.

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.
2x - y + 4z = -2
3x + 2y - z = -3
x + 4y - 2z = 17

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Key Concepts

Cramer's Rule

Determinant of a Matrix

Alternative Methods for Solving Systems