Linear Algebra Examples

$6 x + 3 y = 9$ , $y = - 2 x + 5$

Step 1

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

$[\begin{array}{c} 6 & 3 \\ 2 & 1 \end{array}] \cdot [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 9 \\ 5 \end{array}]$

Step 2

Find the inverse of the coefficient matrix.

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The inverse of a $2 \times 2$ matrix can be found using the formula $\frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$ where $| A |$ is the determinant of $A$ .

If $A = [\begin{array}{c} a & b \\ c & d \end{array}]$ then $A^{- 1} = \frac{1}{| A |} [\begin{array}{c} d & - b \\ - c & a \end{array}]$

Find the determinant of $[\begin{array}{c} 6 & 3 \\ 2 & 1 \end{array}]$ .

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These are both valid notations for the determinant of a matrix.

$determinant [\begin{array}{c} 6 & 3 \\ 2 & 1 \end{array}] = | \begin{array}{c} 6 & 3 \\ 2 & 1 \end{array} |$

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$(6) (1) - 2 \cdot 3$

Simplify the determinant.

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Simplify each term.

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Multiply $6$ by $1$ .

$6 - 2 \cdot 3$

Multiply $- 2$ by $3$ .

$6 - 6$

Subtract $6$ from $6$ .

$0$

Substitute the known values into the formula for the inverse of a matrix.

$\frac{1}{0} [\begin{array}{c} 1 & - (3) \\ - (2) & 6 \end{array}]$

Simplify each element in the matrix.

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Rearrange $- (3)$ .

$\frac{1}{0} [\begin{array}{c} 1 & - 3 \\ - (2) & 6 \end{array}]$

Rearrange $- (2)$ .

$\frac{1}{0} [\begin{array}{c} 1 & - 3 \\ - 2 & 6 \end{array}]$

Multiply $\frac{1}{0}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{0} \cdot 1 & \frac{1}{0} \cdot - 3 \\ \frac{1}{0} \cdot - 2 & \frac{1}{0} \cdot 6 \end{array}]$

Rearrange $\frac{1}{0} \cdot 1$ .

$[\begin{array}{c} Undefined & \frac{1}{0} \cdot - 3 \\ \frac{1}{0} \cdot - 2 & \frac{1}{0} \cdot 6 \end{array}]$

Since the matrix is undefined, it cannot be solved.

$Undefined$

Undefined