Linear Algebra Examples

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 2 & - 1 & 0 & 2 \\ 0 & 1 & 2 & - 4 \\ a & b & c & 3 \end{array}]$

Step 1

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

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Step 1.1

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

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 2 - 2 \cdot 1 & - 1 - 2 \cdot 0 & 0 - 2 \cdot 1 & 2 - 2 \cdot - 1 \\ 0 & 1 & 2 & - 4 \\ a & b & c & 3 \end{array}]$

Step 1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & - 1 & - 2 & 4 \\ 0 & 1 & 2 & - 4 \\ a & b & c & 3 \end{array}]$

Step 2

Perform the row operation $R_{4} = R_{4} - a R_{1}$ to make the entry at $4, 1$ a $0$ .

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Step 2.1

Perform the row operation $R_{4} = R_{4} - a R_{1}$ to make the entry at $4, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & - 1 & - 2 & 4 \\ 0 & 1 & 2 & - 4 \\ a - a \cdot 1 & b - a \cdot 0 & c - a \cdot 1 & 3 - a \cdot - 1 \end{array}]$

Step 2.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & - 1 & - 2 & 4 \\ 0 & 1 & 2 & - 4 \\ 0 & b & c - a & 3 + a \end{array}]$

Step 3

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

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Step 3.1

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

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ - 0 & - - 1 & - - 2 & - 1 \cdot 4 \\ 0 & 1 & 2 & - 4 \\ 0 & b & c - a & 3 + a \end{array}]$

Step 3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 2 & - 4 \\ 0 & 1 & 2 & - 4 \\ 0 & b & c - a & 3 + a \end{array}]$

Step 4

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 4.1

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 2 & - 4 \\ 0 - 0 & 1 - 1 & 2 - 2 & - 4 + 4 \\ 0 & b & c - a & 3 + a \end{array}]$

Step 4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 2 & - 4 \\ 0 & 0 & 0 & 0 \\ 0 & b & c - a & 3 + a \end{array}]$

Step 5

Perform the row operation $R_{4} = R_{4} - b R_{2}$ to make the entry at $4, 2$ a $0$ .

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Step 5.1

Perform the row operation $R_{4} = R_{4} - b R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 2 & - 4 \\ 0 & 0 & 0 & 0 \\ 0 - b \cdot 0 & b - b \cdot 1 & c - a - b \cdot 2 & 3 + a - b \cdot - 4 \end{array}]$

Step 5.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 2 & - 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & c - a - 2 b & 3 + a + 4 b \end{array}]$

Step 6

Swap $R_{4}$ with $R_{3}$ to put a nonzero entry at $3, 3$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 2 & - 4 \\ 0 & 0 & c - a - 2 b & 3 + a + 4 b \\ 0 & 0 & 0 & 0 \end{array}]$

Step 7

Multiply each element of $R_{3}$ by $\frac{1}{c - a - 2 b}$ to make the entry at $3, 3$ a $1$ .

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Step 7.1

Multiply each element of $R_{3}$ by $\frac{1}{c - a - 2 b}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 2 & - 4 \\ \frac{0}{c - a - 2 b} & \frac{0}{c - a - 2 b} & \frac{c - a - 2 b}{c - a - 2 b} & \frac{3 + a + 4 b}{c - a - 2 b} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 2 & - 4 \\ 0 & 0 & 1 & \frac{3 + a + 4 b}{c - a - 2 b} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 8

Perform the row operation $R_{2} = R_{2} - 2 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 8.1

Perform the row operation $R_{2} = R_{2} - 2 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 4 - 2 \frac{3 + a + 4 b}{c - a - 2 b} \\ 0 & 0 & 1 & \frac{3 + a + 4 b}{c - a - 2 b} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 8.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 1 & - 1 \\ 0 & 1 & 0 & \frac{2 (a - 2 c - 3)}{c - a - 2 b} \\ 0 & 0 & 1 & \frac{3 + a + 4 b}{c - a - 2 b} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 9

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 9.1

Perform the row operation $R_{1} = R_{1} - R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 0 & 0 - 0 & 1 - 1 & - 1 - \frac{3 + a + 4 b}{c - a - 2 b} \\ 0 & 1 & 0 & \frac{2 (a - 2 c - 3)}{c - a - 2 b} \\ 0 & 0 & 1 & \frac{3 + a + 4 b}{c - a - 2 b} \\ 0 & 0 & 0 & 0 \end{array}]$

Step 9.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - \frac{2 b + c + 3}{c - a - 2 b} \\ 0 & 1 & 0 & \frac{2 (a - 2 c - 3)}{c - a - 2 b} \\ 0 & 0 & 1 & \frac{3 + a + 4 b}{c - a - 2 b} \\ 0 & 0 & 0 & 0 \end{array}]$