Algebra Examples

$[\begin{array}{c} 3 & 2 \\ - 1 & 5 \end{array}] x = [\begin{array}{c} - 10 & - 11 \\ 26 & - 36 \end{array}]$

Step 1

Find the inverse 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 $[\begin{array}{c} 3 & 2 \\ - 1 & 5 \end{array}]$ .

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

$determinant [\begin{array}{c} 3 & 2 \\ - 1 & 5 \end{array}] = | \begin{array}{c} 3 & 2 \\ - 1 & 5 \end{array} |$

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

$(3) (5) + 1 \cdot 2$

Simplify the determinant.

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Simplify each term.

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Multiply $3$ by $5$ .

$15 + 1 \cdot 2$

Multiply $2$ by $1$ .

$15 + 2$

Add $15$ and $2$ .

$17$

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

$\frac{1}{17} [\begin{array}{c} 5 & - (2) \\ - (- 1) & 3 \end{array}]$

Simplify each element in the matrix.

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Rearrange $- (2)$ .

$\frac{1}{17} [\begin{array}{c} 5 & - 2 \\ - (- 1) & 3 \end{array}]$

Rearrange $- (- 1)$ .

$\frac{1}{17} [\begin{array}{c} 5 & - 2 \\ 1 & 3 \end{array}]$

Multiply $\frac{1}{17}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{17} \cdot 5 & \frac{1}{17} \cdot - 2 \\ \frac{1}{17} \cdot 1 & \frac{1}{17} \cdot 3 \end{array}]$

Simplify each element in the matrix.

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Rearrange $\frac{1}{17} \cdot 5$ .

$[\begin{array}{c} \frac{5}{17} & \frac{1}{17} \cdot - 2 \\ \frac{1}{17} \cdot 1 & \frac{1}{17} \cdot 3 \end{array}]$

Rearrange $\frac{1}{17} \cdot - 2$ .

$[\begin{array}{c} \frac{5}{17} & - \frac{2}{17} \\ \frac{1}{17} \cdot 1 & \frac{1}{17} \cdot 3 \end{array}]$

Rearrange $\frac{1}{17} \cdot 1$ .

$[\begin{array}{c} \frac{5}{17} & - \frac{2}{17} \\ \frac{1}{17} & \frac{1}{17} \cdot 3 \end{array}]$

Rearrange $\frac{1}{17} \cdot 3$ .

$[\begin{array}{c} \frac{5}{17} & - \frac{2}{17} \\ \frac{1}{17} & \frac{3}{17} \end{array}]$

Step 2

Assuming that $X$ is the matrix to solve for, multiply the inverse matrix by both sides of the equation.

$[\begin{array}{c} \frac{5}{17} & - \frac{2}{17} \\ \frac{1}{17} & \frac{3}{17} \end{array}] \cdot ([\begin{array}{c} 3 & 2 \\ - 1 & 5 \end{array}] x) = [\begin{array}{c} \frac{5}{17} & - \frac{2}{17} \\ \frac{1}{17} & \frac{3}{17} \end{array}] \cdot [\begin{array}{c} - 10 & - 11 \\ 26 & - 36 \end{array}]$

Step 3

Simplify the left side of the equation.

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Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} \frac{5}{17} \cdot 3 - \frac{2}{17} \cdot - 1 & \frac{5}{17} \cdot 2 - \frac{2}{17} \cdot 5 \\ \frac{1}{17} \cdot 3 + \frac{3}{17} \cdot - 1 & \frac{1}{17} \cdot 2 + \frac{3}{17} \cdot 5 \end{array}] x = [\begin{array}{c} \frac{5}{17} & - \frac{2}{17} \\ \frac{1}{17} & \frac{3}{17} \end{array}] \cdot [\begin{array}{c} - 10 & - 11 \\ 26 & - 36 \end{array}]$

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} \frac{5}{17} & - \frac{2}{17} \\ \frac{1}{17} & \frac{3}{17} \end{array}] \cdot [\begin{array}{c} - 10 & - 11 \\ 26 & - 36 \end{array}]$

Multiplying the identity matrix by any matrix $x$ is matrix $x$ .

$x = [\begin{array}{c} \frac{5}{17} & - \frac{2}{17} \\ \frac{1}{17} & \frac{3}{17} \end{array}] \cdot [\begin{array}{c} - 10 & - 11 \\ 26 & - 36 \end{array}]$

Step 4

Simplify the right side of the equation.

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Multiply each row in the first matrix by each column in the second matrix.

$x = [\begin{array}{c} \frac{5}{17} \cdot - 10 - \frac{2}{17} \cdot 26 & \frac{5}{17} \cdot - 11 - \frac{2}{17} \cdot - 36 \\ \frac{1}{17} \cdot - 10 + \frac{3}{17} \cdot 26 & \frac{1}{17} \cdot - 11 + \frac{3}{17} \cdot - 36 \end{array}]$

Simplify each element of the matrix by multiplying out all the expressions.

$x = [\begin{array}{c} - 6 & 1 \\ 4 & - 7 \end{array}]$

The matrix is in the most simplified form.

$x = [\begin{array}{c} - 6 & 1 \\ 4 & - 7 \end{array}]$