Linear Algebra Examples

$[\begin{array}{c} a & b & 1 & 0 \\ c & d & 0 & 1 \end{array}]$

Step 1

Multiply each element of $R_{1}$ by $\frac{1}{a}$ to make the entry at $1, 1$ a $1$ .

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Step 1.1

Multiply each element of $R_{1}$ by $\frac{1}{a}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{a}{a} & \frac{b}{a} & \frac{1}{a} & \frac{0}{a} \\ c & d & 0 & 1 \end{array}]$

Step 1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{1}{a} & 0 \\ c & d & 0 & 1 \end{array}]$

Step 2

Perform the row operation $R_{2} = R_{2} - c R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{2} = R_{2} - c R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{1}{a} & 0 \\ c - c \cdot 1 & d - c \frac{b}{a} & 0 - c \frac{1}{a} & 1 - c \cdot 0 \end{array}]$

Step 2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{1}{a} & 0 \\ 0 & d - \frac{b c}{a} & - \frac{c}{a} & 1 \end{array}]$

Step 3

Multiply each element of $R_{2}$ by $\frac{1}{d - \frac{b c}{a}}$ to make the entry at $2, 2$ a $1$ .

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Step 3.1

Multiply each element of $R_{2}$ by $\frac{1}{d - \frac{b c}{a}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{1}{a} & 0 \\ \frac{0}{d - \frac{b c}{a}} & \frac{d - \frac{b c}{a}}{d - \frac{b c}{a}} & \frac{- \frac{c}{a}}{d - \frac{b c}{a}} & \frac{1}{d - \frac{b c}{a}} \end{array}]$

Step 3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{1}{a} & 0 \\ 0 & 1 & - \frac{c}{d a - b c} & \frac{a}{d a - b c} \end{array}]$

Step 4

Perform the row operation $R_{1} = R_{1} - \frac{b}{a} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 4.1

Perform the row operation $R_{1} = R_{1} - \frac{b}{a} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{b}{a} \cdot 0 & \frac{b}{a} - \frac{b}{a} \cdot 1 & \frac{1}{a} - \frac{b}{a} (- \frac{c}{d a - b c}) & 0 - \frac{b}{a} \cdot \frac{a}{d a - b c} \\ 0 & 1 & - \frac{c}{d a - b c} & \frac{a}{d a - b c} \end{array}]$

Step 4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & \frac{1}{a} + \frac{b c}{a (d a - b c)} & - \frac{b}{d a - b c} \\ 0 & 1 & - \frac{c}{d a - b c} & \frac{a}{d a - b c} \end{array}]$