Solve the Matrix Equation [[2,-1],[7,-4]]x=[[10,-3],[39,-7]]

$[\begin{array}{c} 2 & - 1 \\ 7 & - 4 \end{array}] x = [\begin{array}{c} 10 & - 3 \\ 39 & - 7 \end{array}]$

Step 1

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 $[\begin{array}{c} 2 & - 1 \\ 7 & - 4 \end{array}]$ .

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

$determinant [\begin{array}{c} 2 & - 1 \\ 7 & - 4 \end{array}] = | \begin{array}{c} 2 & - 1 \\ 7 & - 4 \end{array} |$

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

$(2) (- 4) - 7 \cdot - 1$

Simplify the determinant.

Simplify each term.

Multiply $2$ by $- 4$ .

$- 8 - 7 \cdot - 1$

Multiply $- 7$ by $- 1$ .

$- 8 + 7$

Add $- 8$ and $7$ .

$- 1$

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

$\frac{1}{- 1} [\begin{array}{c} - 4 & - (- 1) \\ - (7) & 2 \end{array}]$

Simplify each element in the matrix.

Rearrange $- (- 1)$ .

$\frac{1}{- 1} [\begin{array}{c} - 4 & 1 \\ - (7) & 2 \end{array}]$

Rearrange $- (7)$ .

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

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

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

Simplify each element in the matrix.

Rearrange $\frac{1}{- 1} \cdot - 4$ .

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

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

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

Rearrange $\frac{1}{- 1} \cdot - 7$ .

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

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

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

Step 2

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

$[\begin{array}{c} 4 & - 1 \\ 7 & - 2 \end{array}] \cdot ([\begin{array}{c} 2 & - 1 \\ 7 & - 4 \end{array}] x) = [\begin{array}{c} 4 & - 1 \\ 7 & - 2 \end{array}] \cdot [\begin{array}{c} 10 & - 3 \\ 39 & - 7 \end{array}]$

Step 3

Simplify the left side of the equation.

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

$[\begin{array}{c} 4 \cdot 2 - 1 \cdot 7 & 4 \cdot - 1 - 1 \cdot - 4 \\ 7 \cdot 2 - 2 \cdot 7 & 7 \cdot - 1 - 2 \cdot - 4 \end{array}] x = [\begin{array}{c} 4 & - 1 \\ 7 & - 2 \end{array}] \cdot [\begin{array}{c} 10 & - 3 \\ 39 & - 7 \end{array}]$

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

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \end{array}] x = [\begin{array}{c} 4 & - 1 \\ 7 & - 2 \end{array}] \cdot [\begin{array}{c} 10 & - 3 \\ 39 & - 7 \end{array}]$

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

$x = [\begin{array}{c} 4 & - 1 \\ 7 & - 2 \end{array}] \cdot [\begin{array}{c} 10 & - 3 \\ 39 & - 7 \end{array}]$

Step 4

Simplify the right side of the equation.

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

$x = [\begin{array}{c} 4 \cdot 10 - 1 \cdot 39 & 4 \cdot - 3 - 1 \cdot - 7 \\ 7 \cdot 10 - 2 \cdot 39 & 7 \cdot - 3 - 2 \cdot - 7 \end{array}]$

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

$x = [\begin{array}{c} 1 & - 5 \\ - 8 & - 7 \end{array}]$

The matrix is in the most simplified form.

$x = [\begin{array}{c} 1 & - 5 \\ - 8 & - 7 \end{array}]$

Solve the Matrix Equation [[2,-1],[7,-4]]x=[[10,-3],[39,-7]]

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