Linear Algebra Examples

$[\begin{array}{c} 2 & 4 \\ 6 & 8 \end{array}] - [\begin{array}{c} 1 & 2 \\ 3 & 4 \end{array}]$

Combine the similar matrices with each others.

$[\begin{array}{c} 2 & 4 \\ 6 & 8 \end{array}] - [\begin{array}{c} 1 & 2 \\ 3 & 4 \end{array}]$

Simplify each element of the matrix $[\begin{array}{c} 2 - 1 & 4 - 2 \\ 6 - 3 & 8 - 4 \end{array}]$ .

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Subtract $[\begin{array}{c} 1 & 2 \\ 3 & 4 \end{array}]$ from $[\begin{array}{c} 2 & 4 \\ 6 & 8 \end{array}]$ by subtracting the corresponding elements of $[\begin{array}{c} 1 & 2 \\ 3 & 4 \end{array}]$ from each element of $[\begin{array}{c} 2 & 4 \\ 6 & 8 \end{array}]$ .

$[\begin{array}{c} 2 - 1 & 4 - 2 \\ 6 - 3 & 8 - 4 \end{array}]$

Simplify element $0, 0$ of the matrix.

$[\begin{array}{c} 1 & 4 - 2 \\ 6 - 3 & 8 - 4 \end{array}]$

Simplify element $0, 1$ of the matrix.

$[\begin{array}{c} 1 & 2 \\ 6 - 3 & 8 - 4 \end{array}]$

Simplify element $1, 0$ of the matrix.

$[\begin{array}{c} 1 & 2 \\ 3 & 8 - 4 \end{array}]$

Simplify element $1, 1$ of the matrix.

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

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

The determinant of $[\begin{array}{c} 1 & 2 \\ 3 & 4 \end{array}]$ is $- 2$ .

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

$determinant [\begin{array}{c} 1 & 2 \\ 3 & 4 \end{array}] = | \begin{array}{c} 1 & 2 \\ 3 & 4 \end{array} |$

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

$(1) (4) - 3 \cdot 2$

Simplify the determinant.

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Simplify each term.

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Multiply $4$ by $1$ .

$4 - 3 \cdot 2$

Multiply $- 3$ by $2$ .

$4 - 6$

Subtract $6$ from $4$ .

$- 2$

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

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

Simplify each element of the matrix $[\begin{array}{c} 4 & - (2) \\ - (3) & 1 \end{array}]$ .

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Simplify element $0, 1$ by multiplying $- (2)$ .

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

Simplify element $1, 0$ by multiplying $- (3)$ .

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

Multiply $\frac{1}{- 2}$ by each element of the matrix.

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

Simplify each element of the matrix $[\begin{array}{c} \frac{1}{- 2} \cdot 4 & \frac{1}{- 2} \cdot - 2 \\ \frac{1}{- 2} \cdot - 3 & \frac{1}{- 2} \cdot 1 \end{array}]$ .

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Simplify element $0, 0$ by multiplying $\frac{1}{- 2} \cdot 4$ .

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

Simplify element $0, 1$ by multiplying $\frac{1}{- 2} \cdot - 2$ .

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

Simplify element $1, 0$ by multiplying $\frac{1}{- 2} \cdot - 3$ .

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

Simplify element $1, 1$ by multiplying $\frac{1}{- 2} \cdot 1$ .

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

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