Linear Algebra Examples

$3 x + y + 2 z = - 6$ , $2 x - 3 y - z = - 3$ , $x + 4 y + 3 z = - 3$

Step 1

Find the $A X = B$ from the system of equations.

$[\begin{array}{c} 3 & 1 & 2 \\ 2 & - 3 & - 1 \\ 1 & 4 & 3 \end{array}] \cdot [\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} - 6 \\ - 3 \\ - 3 \end{array}]$

Step 2

Find the inverse of the coefficient matrix.

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Set up a matrix that is broken into two pieces of equal size. On the left side, fill in the elements of the original matrix. On the right side, fill in elements of the identity matrix. In order to find the inverse matrix, use row operations to convert the left side into the identity matrix. After this is complete, the inverse of the original matrix will be on the right side of the double matrix.

$[\begin{array}{c} 3 & 1 & 2 & 1 & 0 & 0 \\ 2 & - 3 & - 1 & 0 & 1 & 0 \\ 1 & 4 & 3 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{1} = \frac{1}{3} R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $1$ .

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Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = \frac{1}{3} R_{1}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} \frac{1}{3} R_{1} & \frac{1}{3} R_{1} & \frac{1}{3} R_{1} & \frac{1}{3} R_{1} & \frac{1}{3} R_{1} & \frac{1}{3} R_{1} \\ 2 & - 3 & - 1 & 0 & 1 & 0 \\ 1 & 4 & 3 & 0 & 0 & 1 \end{array}]$

$R_{1} = \frac{1}{3} R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = \frac{1}{3} R_{1}$ .

$[\begin{array}{c} (\frac{1}{3}) \cdot (3) & (\frac{1}{3}) \cdot (1) & (\frac{1}{3}) \cdot (2) & (\frac{1}{3}) \cdot (1) & (\frac{1}{3}) \cdot (0) & (\frac{1}{3}) \cdot (0) \\ 2 & - 3 & - 1 & 0 & 1 & 0 \\ 1 & 4 & 3 & 0 & 0 & 1 \end{array}]$

$R_{1} = \frac{1}{3} R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ 2 & - 3 & - 1 & 0 & 1 & 0 \\ 1 & 4 & 3 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{2} = - 2 \cdot R_{1} + R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - 2 \cdot R_{1} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ - 2 \cdot R_{1} + R_{2} & - 2 \cdot R_{1} + R_{2} & - 2 \cdot R_{1} + R_{2} & - 2 \cdot R_{1} + R_{2} & - 2 \cdot R_{1} + R_{2} & - 2 \cdot R_{1} + R_{2} \\ 1 & 4 & 3 & 0 & 0 & 1 \end{array}]$

$R_{2} = - 2 \cdot R_{1} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 2 \cdot R_{1} + R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ (- 2) \cdot (1) + 2 & (- 2) \cdot (\frac{1}{3}) - 3 & (- 2) \cdot (\frac{2}{3}) - 1 & (- 2) \cdot (\frac{1}{3}) + 0 & (- 2) \cdot (0) + 1 & (- 2) \cdot (0) + 0 \\ 1 & 4 & 3 & 0 & 0 & 1 \end{array}]$

$R_{2} = - 2 \cdot R_{1} + R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ 0 & - \frac{11}{3} & - \frac{7}{3} & - \frac{2}{3} & 1 & 0 \\ 1 & 4 & 3 & 0 & 0 & 1 \end{array}]$

Perform the row operation $R_{3} = - 1 \cdot R_{1} + R_{3}$ on $R_{3}$ (row $3$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{3}$ (row $3$ ) with the row operation $R_{3} = - 1 \cdot R_{1} + R_{3}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ 0 & - \frac{11}{3} & - \frac{7}{3} & - \frac{2}{3} & 1 & 0 \\ - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} & - 1 \cdot R_{1} + R_{3} \end{array}]$

$R_{3} = - 1 \cdot R_{1} + R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - 1 \cdot R_{1} + R_{3}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ 0 & - \frac{11}{3} & - \frac{7}{3} & - \frac{2}{3} & 1 & 0 \\ (- 1) \cdot (1) + 1 & (- 1) \cdot (\frac{1}{3}) + 4 & (- 1) \cdot (\frac{2}{3}) + 3 & (- 1) \cdot (\frac{1}{3}) + 0 & (- 1) \cdot (0) + 0 & (- 1) \cdot (0) + 1 \end{array}]$

$R_{3} = - 1 \cdot R_{1} + R_{3}$

Simplify $R_{3}$ (row $3$ ).

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ 0 & - \frac{11}{3} & - \frac{7}{3} & - \frac{2}{3} & 1 & 0 \\ 0 & \frac{11}{3} & \frac{7}{3} & - \frac{1}{3} & 0 & 1 \end{array}]$

Perform the row operation $R_{2} = - \frac{3}{11} R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $1$ .

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Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - \frac{3}{11} R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ - \frac{3}{11} R_{2} & - \frac{3}{11} R_{2} & - \frac{3}{11} R_{2} & - \frac{3}{11} R_{2} & - \frac{3}{11} R_{2} & - \frac{3}{11} R_{2} \\ 0 & \frac{11}{3} & \frac{7}{3} & - \frac{1}{3} & 0 & 1 \end{array}]$

$R_{2} = - \frac{3}{11} R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - \frac{3}{11} R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ (- \frac{3}{11}) \cdot (0) & (- \frac{3}{11}) \cdot (- \frac{11}{3}) & (- \frac{3}{11}) \cdot (- \frac{7}{3}) & (- \frac{3}{11}) \cdot (- \frac{2}{3}) & (- \frac{3}{11}) \cdot (1) & (- \frac{3}{11}) \cdot (0) \\ 0 & \frac{11}{3} & \frac{7}{3} & - \frac{1}{3} & 0 & 1 \end{array}]$

$R_{2} = - \frac{3}{11} R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & \frac{1}{3} & \frac{2}{3} & \frac{1}{3} & 0 & 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & \frac{11}{3} & \frac{7}{3} & - \frac{1}{3} & 0 & 1 \end{array}]$

Perform the row operation $R_{1} = - \frac{1}{3} R_{2} + R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = - \frac{1}{3} R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - \frac{1}{3} R_{2} + R_{1} & - \frac{1}{3} R_{2} + R_{1} & - \frac{1}{3} R_{2} + R_{1} & - \frac{1}{3} R_{2} + R_{1} & - \frac{1}{3} R_{2} + R_{1} & - \frac{1}{3} R_{2} + R_{1} \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & \frac{11}{3} & \frac{7}{3} & - \frac{1}{3} & 0 & 1 \end{array}]$

$R_{1} = - \frac{1}{3} R_{2} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - \frac{1}{3} R_{2} + R_{1}$ .

$[\begin{array}{c} (- \frac{1}{3}) \cdot (0) + 1 & (- \frac{1}{3}) \cdot (1) + \frac{1}{3} & (- \frac{1}{3}) \cdot (\frac{7}{11}) + \frac{2}{3} & (- \frac{1}{3}) \cdot (\frac{2}{11}) + \frac{1}{3} & (- \frac{1}{3}) \cdot (- \frac{3}{11}) + 0 & (- \frac{1}{3}) \cdot (0) + 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & \frac{11}{3} & \frac{7}{3} & - \frac{1}{3} & 0 & 1 \end{array}]$

$R_{1} = - \frac{1}{3} R_{2} + R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & \frac{3}{11} & \frac{1}{11} & 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & \frac{11}{3} & \frac{7}{3} & - \frac{1}{3} & 0 & 1 \end{array}]$

Perform the row operation $R_{3} = - \frac{11}{3} R_{2} + R_{3}$ on $R_{3}$ (row $3$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{3}$ (row $3$ ) with the row operation $R_{3} = - \frac{11}{3} R_{2} + R_{3}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & \frac{3}{11} & \frac{1}{11} & 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ - \frac{11}{3} R_{2} + R_{3} & - \frac{11}{3} R_{2} + R_{3} & - \frac{11}{3} R_{2} + R_{3} & - \frac{11}{3} R_{2} + R_{3} & - \frac{11}{3} R_{2} + R_{3} & - \frac{11}{3} R_{2} + R_{3} \end{array}]$

$R_{3} = - \frac{11}{3} R_{2} + R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - \frac{11}{3} R_{2} + R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & \frac{3}{11} & \frac{1}{11} & 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ (- \frac{11}{3}) \cdot (0) + 0 & (- \frac{11}{3}) \cdot (1) + \frac{11}{3} & (- \frac{11}{3}) \cdot (\frac{7}{11}) + \frac{7}{3} & (- \frac{11}{3}) \cdot (\frac{2}{11}) - \frac{1}{3} & (- \frac{11}{3}) \cdot (- \frac{3}{11}) + 0 & (- \frac{11}{3}) \cdot (0) + 1 \end{array}]$

$R_{3} = - \frac{11}{3} R_{2} + R_{3}$

Simplify $R_{3}$ (row $3$ ).

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & \frac{3}{11} & \frac{1}{11} & 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & 0 & 0 & - 1 & 1 & 1 \end{array}]$

Perform the row operation $R_{3} = - 1 \cdot R_{3}$ on $R_{3}$ (row $3$ ) in order to convert some elements in the row to $1$ .

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Replace $R_{3}$ (row $3$ ) with the row operation $R_{3} = - 1 \cdot R_{3}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & \frac{3}{11} & \frac{1}{11} & 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ - 1 \cdot R_{3} & - 1 \cdot R_{3} & - 1 \cdot R_{3} & - 1 \cdot R_{3} & - 1 \cdot R_{3} & - 1 \cdot R_{3} \end{array}]$

$R_{3} = - 1 \cdot R_{3}$

Replace $R_{3}$ (row $3$ ) with the actual values of the elements for the row operation $R_{3} = - 1 \cdot R_{3}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & \frac{3}{11} & \frac{1}{11} & 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ (- 1) \cdot (0) & (- 1) \cdot (0) & (- 1) \cdot (0) & (- 1) \cdot (- 1) & (- 1) \cdot (1) & (- 1) \cdot (1) \end{array}]$

$R_{3} = - 1 \cdot R_{3}$

Simplify $R_{3}$ (row $3$ ).

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & \frac{3}{11} & \frac{1}{11} & 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 \end{array}]$

Perform the row operation $R_{1} = - \frac{3}{11} R_{3} + R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = - \frac{3}{11} R_{3} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - \frac{3}{11} R_{3} + R_{1} & - \frac{3}{11} R_{3} + R_{1} & - \frac{3}{11} R_{3} + R_{1} & - \frac{3}{11} R_{3} + R_{1} & - \frac{3}{11} R_{3} + R_{1} & - \frac{3}{11} R_{3} + R_{1} \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 \end{array}]$

$R_{1} = - \frac{3}{11} R_{3} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - \frac{3}{11} R_{3} + R_{1}$ .

$[\begin{array}{c} (- \frac{3}{11}) \cdot (0) + 1 & (- \frac{3}{11}) \cdot (0) + 0 & (- \frac{3}{11}) \cdot (0) + \frac{5}{11} & (- \frac{3}{11}) \cdot (1) + \frac{3}{11} & (- \frac{3}{11}) \cdot (- 1) + \frac{1}{11} & (- \frac{3}{11}) \cdot (- 1) + 0 \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 \end{array}]$

$R_{1} = - \frac{3}{11} R_{3} + R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & 0 & \frac{4}{11} & \frac{3}{11} \\ 0 & 1 & \frac{7}{11} & \frac{2}{11} & - \frac{3}{11} & 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 \end{array}]$

Perform the row operation $R_{2} = - \frac{2}{11} R_{3} + R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - \frac{2}{11} R_{3} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & 0 & \frac{4}{11} & \frac{3}{11} \\ - \frac{2}{11} R_{3} + R_{2} & - \frac{2}{11} R_{3} + R_{2} & - \frac{2}{11} R_{3} + R_{2} & - \frac{2}{11} R_{3} + R_{2} & - \frac{2}{11} R_{3} + R_{2} & - \frac{2}{11} R_{3} + R_{2} \\ 0 & 0 & 0 & 1 & - 1 & - 1 \end{array}]$

$R_{2} = - \frac{2}{11} R_{3} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - \frac{2}{11} R_{3} + R_{2}$ .

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & 0 & \frac{4}{11} & \frac{3}{11} \\ (- \frac{2}{11}) \cdot (0) + 0 & (- \frac{2}{11}) \cdot (0) + 1 & (- \frac{2}{11}) \cdot (0) + \frac{7}{11} & (- \frac{2}{11}) \cdot (1) + \frac{2}{11} & (- \frac{2}{11}) \cdot (- 1) - \frac{3}{11} & (- \frac{2}{11}) \cdot (- 1) + 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 \end{array}]$

$R_{2} = - \frac{2}{11} R_{3} + R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & 0 & \frac{5}{11} & 0 & \frac{4}{11} & \frac{3}{11} \\ 0 & 1 & \frac{7}{11} & 0 & - \frac{1}{11} & \frac{2}{11} \\ 0 & 0 & 0 & 1 & - 1 & - 1 \end{array}]$

Since the determinant of the matrix is zero, there is no inverse.

No inverse

Step 3

Since the matrix has no inverse, it cannot be solved using the inverse matrix.

No solution