Lineare Algebra Beispiele

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 4 & y 4.4 & y 4.2 & y 4.1 & y x & 0.7 x & 0.6 x & 0.1 x & 9 x & 6 x & 7 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 4 & y \\ 4.4 & y \\ 4.2 & y \\ 4.1 & y \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1

Ermittele die normierte Zeilenstufenform.

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Schritt 1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{4}$ to make the entry at

$1, 1$ a

$1$ .

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Schritt 1.1.1

Multiply each element of

$R_{1}$ by

$\frac{1}{4}$ to make the entry at

$1, 1$ a

$1$ .

$[\begin{array}{c} \frac{4}{4} & \frac{y}{4} \\ 4.4 & y \\ 4.2 & y \\ 4.1 & y \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.1.2

Vereinfache

$R_{1}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 4.4 & y \\ 4.2 & y \\ 4.1 & y \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.2

Perform the row operation

$R_{2} = R_{2} - 4.4 R_{1}$ to make the entry at

$2, 1$ a

$0$ .

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Schritt 1.2.1

Perform the row operation

$R_{2} = R_{2} - 4.4 R_{1}$ to make the entry at

$2, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 4.4 - 4.4 \cdot 1 & y - 4.4 \frac{y}{4} \\ 4.2 & y \\ 4.1 & y \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.2.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 4.2 & y \\ 4.1 & y \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.3

Perform the row operation

$R_{3} = R_{3} - 4.2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

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Schritt 1.3.1

Perform the row operation

$R_{3} = R_{3} - 4.2 R_{1}$ to make the entry at

$3, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 4.2 - 4.2 \cdot 1 & y - 4.2 \frac{y}{4} \\ 4.1 & y \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.3.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 4.1 & y \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.4

Perform the row operation

$R_{4} = R_{4} - 4.1 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

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Schritt 1.4.1

Perform the row operation

$R_{4} = R_{4} - 4.1 R_{1}$ to make the entry at

$4, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 4.1 - 4.1 \cdot 1 & y - 4.1 \frac{y}{4} \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.4.2

Vereinfache

$R_{4}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ x & 0.7 \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.5

Perform the row operation

$R_{5} = R_{5} - x R_{1}$ to make the entry at

$5, 1$ a

$0$ .

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Schritt 1.5.1

Perform the row operation

$R_{5} = R_{5} - x R_{1}$ to make the entry at

$5, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ x - x \cdot 1 & 0.7 - x \frac{y}{4} \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.5.2

Vereinfache

$R_{5}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ x & 0.6 \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.6

Perform the row operation

$R_{6} = R_{6} - x R_{1}$ to make the entry at

$6, 1$ a

$0$ .

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Schritt 1.6.1

Perform the row operation

$R_{6} = R_{6} - x R_{1}$ to make the entry at

$6, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ x - x \cdot 1 & 0.6 - x \frac{y}{4} \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.6.2

Vereinfache

$R_{6}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ x & 0.1 \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.7

Perform the row operation

$R_{7} = R_{7} - x R_{1}$ to make the entry at

$7, 1$ a

$0$ .

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Schritt 1.7.1

Perform the row operation

$R_{7} = R_{7} - x R_{1}$ to make the entry at

$7, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ x - x \cdot 1 & 0.1 - x \frac{y}{4} \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.7.2

Vereinfache

$R_{7}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ x & 9 \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.8

Perform the row operation

$R_{8} = R_{8} - x R_{1}$ to make the entry at

$8, 1$ a

$0$ .

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Schritt 1.8.1

Perform the row operation

$R_{8} = R_{8} - x R_{1}$ to make the entry at

$8, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ x - x \cdot 1 & 9 - x \frac{y}{4} \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.8.2

Vereinfache

$R_{8}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ x & 6 \\ x & 7 \end{array}]$

Schritt 1.9

Perform the row operation

$R_{9} = R_{9} - x R_{1}$ to make the entry at

$9, 1$ a

$0$ .

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Schritt 1.9.1

Perform the row operation

$R_{9} = R_{9} - x R_{1}$ to make the entry at

$9, 1$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ x - x \cdot 1 & 6 - x \frac{y}{4} \\ x & 7 \end{array}]$

Schritt 1.9.2

Vereinfache

$R_{9}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & - 0.1 y \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ x & 7 \end{array}]$

Schritt 1.10

Perform the row operation

$R_{10} = R_{10} - x R_{1}$ to make the entry at

$10, 1$ a

$0$ .

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Schritt 1.10.1

Perform the row operation

$R_{10} = R_{10} - x R_{1}$ to make the entry at

$10, 1$ a

$0$ .

Schritt 1.10.2

Vereinfache

$R_{10}$ .

Schritt 1.11

Multiply each element of

$R_{2}$ by

$\frac{1}{- 0.1 y}$ to make the entry at

$2, 2$ a

$1$ .

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Schritt 1.11.1

Multiply each element of

$R_{2}$ by

$\frac{1}{- 0.1 y}$ to make the entry at

$2, 2$ a

$1$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ \frac{0}{- 0.1 y} & \frac{- 0.1 y}{- 0.1 y} \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.11.2

Vereinfache

$R_{2}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & - 0.05 y \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.12

Perform the row operation

$R_{3} = R_{3} + 0.05 y R_{2}$ to make the entry at

$3, 2$ a

$0$ .

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Schritt 1.12.1

Perform the row operation

$R_{3} = R_{3} + 0.05 y R_{2}$ to make the entry at

$3, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 + 0.05 y \cdot 0 & - 0.05 y + 0.05 y \cdot 1 \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.12.2

Vereinfache

$R_{3}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & - 0.025 y \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.13

Perform the row operation

$R_{4} = R_{4} + 0.025 y R_{2}$ to make the entry at

$4, 2$ a

$0$ .

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Schritt 1.13.1

Perform the row operation

$R_{4} = R_{4} + 0.025 y R_{2}$ to make the entry at

$4, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 + 0.025 y \cdot 0 & - 0.025 y + 0.025 y \cdot 1 \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.13.2

Vereinfache

$R_{4}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0.7 - \frac{y x}{4} \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.14

Perform the row operation

$R_{5} = R_{5} - (0.7 - \frac{y x}{4}) R_{2}$ to make the entry at

$5, 2$ a

$0$ .

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Schritt 1.14.1

Perform the row operation

$R_{5} = R_{5} - (0.7 - \frac{y x}{4}) R_{2}$ to make the entry at

$5, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 - (0.7 - \frac{y x}{4}) \cdot 0 & 0.7 - \frac{y x}{4} - (0.7 - \frac{y x}{4}) \cdot 1 \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.14.2

Vereinfache

$R_{5}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0.6 - \frac{y x}{4} \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.15

Perform the row operation

$R_{6} = R_{6} - (0.6 - \frac{y x}{4}) R_{2}$ to make the entry at

$6, 2$ a

$0$ .

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Schritt 1.15.1

Perform the row operation

$R_{6} = R_{6} - (0.6 - \frac{y x}{4}) R_{2}$ to make the entry at

$6, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 - (0.6 - \frac{y x}{4}) \cdot 0 & 0.6 - \frac{y x}{4} - (0.6 - \frac{y x}{4}) \cdot 1 \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.15.2

Vereinfache

$R_{6}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0.1 - \frac{y x}{4} \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.16

Perform the row operation

$R_{7} = R_{7} - (0.1 - \frac{y x}{4}) R_{2}$ to make the entry at

$7, 2$ a

$0$ .

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Schritt 1.16.1

Perform the row operation

$R_{7} = R_{7} - (0.1 - \frac{y x}{4}) R_{2}$ to make the entry at

$7, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 - (0.1 - \frac{y x}{4}) \cdot 0 & 0.1 - \frac{y x}{4} - (0.1 - \frac{y x}{4}) \cdot 1 \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.16.2

Vereinfache

$R_{7}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 9 - \frac{y x}{4} \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.17

Perform the row operation

$R_{8} = R_{8} - (9 - \frac{y x}{4}) R_{2}$ to make the entry at

$8, 2$ a

$0$ .

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Schritt 1.17.1

Perform the row operation

$R_{8} = R_{8} - (9 - \frac{y x}{4}) R_{2}$ to make the entry at

$8, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 - (9 - \frac{y x}{4}) \cdot 0 & 9 - \frac{y x}{4} - (9 - \frac{y x}{4}) \cdot 1 \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.17.2

Vereinfache

$R_{8}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 6 - \frac{y x}{4} \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.18

Perform the row operation

$R_{9} = R_{9} - (6 - \frac{y x}{4}) R_{2}$ to make the entry at

$9, 2$ a

$0$ .

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Schritt 1.18.1

Perform the row operation

$R_{9} = R_{9} - (6 - \frac{y x}{4}) R_{2}$ to make the entry at

$9, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 - (6 - \frac{y x}{4}) \cdot 0 & 6 - \frac{y x}{4} - (6 - \frac{y x}{4}) \cdot 1 \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.18.2

Vereinfache

$R_{9}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 7 - \frac{y x}{4} \end{array}]$

Schritt 1.19

Perform the row operation

$R_{10} = R_{10} - (7 - \frac{y x}{4}) R_{2}$ to make the entry at

$10, 2$ a

$0$ .

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Schritt 1.19.1

Perform the row operation

$R_{10} = R_{10} - (7 - \frac{y x}{4}) R_{2}$ to make the entry at

$10, 2$ a

$0$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 - (7 - \frac{y x}{4}) \cdot 0 & 7 - \frac{y x}{4} - (7 - \frac{y x}{4}) \cdot 1 \end{array}]$

Schritt 1.19.2

Vereinfache

$R_{10}$ .

$[\begin{array}{c} 1 & \frac{y}{4} \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}]$

Schritt 1.20

Perform the row operation

$R_{1} = R_{1} - \frac{y}{4} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

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Schritt 1.20.1

Perform the row operation

$R_{1} = R_{1} - \frac{y}{4} R_{2}$ to make the entry at

$1, 2$ a

$0$ .

$[\begin{array}{c} 1 - \frac{y}{4} \cdot 0 & \frac{y}{4} - \frac{y}{4} \cdot 1 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}]$

Schritt 1.20.2

Vereinfache

$R_{1}$ .

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

Schritt 2

The pivot positions are the locations with the leading

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

$a_{11}$ and

$a_{22}$

Pivot Columns:

$1$ and

$2$