Solve the Matrix Equation [[-1,-2],[2,9]]b=[[-3,-5,13],[21,0,-36]]

$[\begin{array}{c} - 1 & - 2 \\ 2 & 9 \end{array}] b = [\begin{array}{c} - 3 & - 5 & 13 \\ 21 & 0 & - 36 \end{array}]$

Find the inverse matrix.

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 $[\begin{array}{c} - 1 & - 2 \\ 2 & 9 \end{array}]$ .

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

$determinant [\begin{array}{c} - 1 & - 2 \\ 2 & 9 \end{array}] = | \begin{array}{c} - 1 & - 2 \\ 2 & 9 \end{array} |$

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

$(- 1) (9) - 2 \cdot - 2$

Simplify the determinant.

Simplify each term.

Multiply $- 1$ by $9$ .

$- 9 - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$- 9 + 4$

Add $- 9$ and $4$ .

$- 5$

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

$\frac{1}{- 5} [\begin{array}{c} 9 & - (- 2) \\ - (2) & - 1 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 2)$ .

$\frac{1}{- 5} [\begin{array}{c} 9 & 2 \\ - (2) & - 1 \end{array}]$

Rearrange $- (2)$ .

$\frac{1}{- 5} [\begin{array}{c} 9 & 2 \\ - 2 & - 1 \end{array}]$

Multiply $\frac{1}{- 5}$ by each element of the matrix.

$[\begin{array}{c} \frac{1}{- 5} \cdot 9 & \frac{1}{- 5} \cdot 2 \\ \frac{1}{- 5} \cdot - 2 & \frac{1}{- 5} \cdot - 1 \end{array}]$

Simplify each element in the matrix.

Rearrange $\frac{1}{- 5} \cdot 9$ .

$[\begin{array}{c} - \frac{9}{5} & \frac{1}{- 5} \cdot 2 \\ \frac{1}{- 5} \cdot - 2 & \frac{1}{- 5} \cdot - 1 \end{array}]$

Rearrange $\frac{1}{- 5} \cdot 2$ .

$[\begin{array}{c} - \frac{9}{5} & - \frac{2}{5} \\ \frac{1}{- 5} \cdot - 2 & \frac{1}{- 5} \cdot - 1 \end{array}]$

Rearrange $\frac{1}{- 5} \cdot - 2$ .

$[\begin{array}{c} - \frac{9}{5} & - \frac{2}{5} \\ \frac{2}{5} & \frac{1}{- 5} \cdot - 1 \end{array}]$

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

$[\begin{array}{c} - \frac{9}{5} & - \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} \end{array}]$

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

$[\begin{array}{c} - \frac{9}{5} & - \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} \end{array}] \cdot ([\begin{array}{c} - 1 & - 2 \\ 2 & 9 \end{array}] x) = [\begin{array}{c} - \frac{9}{5} & - \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} \end{array}] \cdot [\begin{array}{c} - 3 & - 5 & 13 \\ 21 & 0 & - 36 \end{array}]$

Simplify the left side of the equation.

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

$[\begin{array}{c} - \frac{9}{5} \cdot - 1 - \frac{2}{5} \cdot 2 & - \frac{9}{5} \cdot - 2 - \frac{2}{5} \cdot 9 \\ \frac{2}{5} \cdot - 1 + \frac{1}{5} \cdot 2 & \frac{2}{5} \cdot - 2 + \frac{1}{5} \cdot 9 \end{array}] x = [\begin{array}{c} - \frac{9}{5} & - \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} \end{array}] \cdot [\begin{array}{c} - 3 & - 5 & 13 \\ 21 & 0 & - 36 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} - \frac{9}{5} & - \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} \end{array}] \cdot [\begin{array}{c} - 3 & - 5 & 13 \\ 21 & 0 & - 36 \end{array}]$

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

$x = [\begin{array}{c} - \frac{9}{5} & - \frac{2}{5} \\ \frac{2}{5} & \frac{1}{5} \end{array}] \cdot [\begin{array}{c} - 3 & - 5 & 13 \\ 21 & 0 & - 36 \end{array}]$

Simplify the right side of the equation.

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

$x = [\begin{array}{c} - \frac{9}{5} \cdot - 3 - \frac{2}{5} \cdot 21 & - \frac{9}{5} \cdot - 5 - \frac{2}{5} \cdot 0 & - \frac{9}{5} \cdot 13 - \frac{2}{5} \cdot - 36 \\ \frac{2}{5} \cdot - 3 + \frac{1}{5} \cdot 21 & \frac{2}{5} \cdot - 5 + \frac{1}{5} \cdot 0 & \frac{2}{5} \cdot 13 + \frac{1}{5} \cdot - 36 \end{array}]$

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

$x = [\begin{array}{c} - 3 & 9 & - 9 \\ 3 & - 2 & - 2 \end{array}]$

The matrix is in the most simplified form.

$x = [\begin{array}{c} - 3 & 9 & - 9 \\ 3 & - 2 & - 2 \end{array}]$

Solve the Matrix Equation [[-1,-2],[2,9]]b=[[-3,-5,13],[21,0,-36]]

Related Questions

Support LCMGCF.com by disabling your adblocker.