Linear Algebra Examples

$y = C + I + G + (a - M)$ , $y = 51$ , $C = 10 i$ , $G = 550$ , $a = 52$ , $M = 32 I$ , $I = 25$ , $i = 30$

Step 1

Remove parentheses.

$y = C + I + G + a - M, y = 51, C = 10 i, G = 550, a = 52, M = 32 I, I = 25, i = 30$

Step 2

Move all terms containing variables to the left side of the equation.

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Step 2.1

Subtract $C$ from both sides of the equation.

$y - C = I + G + a - M, y = 51, C = 10 i, G = 550, a = 52, M = 32 I, I = 25, i = 30$

Step 2.2

Subtract $I$ from both sides of the equation.

$y - C - I = G + a - M, y = 51, C = 10 i, G = 550, a = 52, M = 32 I, I = 25, i = 30$

Step 2.3

Subtract $G$ from both sides of the equation.

$y - C - I - G = a - M, y = 51, C = 10 i, G = 550, a = 52, M = 32 I, I = 25, i = 30$

Step 2.4

Subtract $a$ from both sides of the equation.

$y - C - I - G - a = - M, y = 51, C = 10 i, G = 550, a = 52, M = 32 I, I = 25, i = 30$

Step 2.5

Add $M$ to both sides of the equation.

$y - C - I - G - a + M = 0, y = 51, C = 10 i, G = 550, a = 52, M = 32 I, I = 25, i = 30$

Step 3

Subtract $32 I$ from both sides of the equation.

$y - C - I - G - a + M = 0, y = 51, C = 10 i, G = 550, a = 52, M - 32 I = 0, I = 25, i = 30$

Step 4

Subtract $i$ from both sides of the equation.

$y - C - I - G - a + M = 0, y = 51, C = 10 i, G = 550, a = 52, M - 32 I = 0, I = 25, 0 = 30 - i$

Step 5

Write the system of equations in matrix form.

$[\begin{array}{c} - 1 & - 1 & - 1 & - 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 1 & 0 & 0 & 0 & 0 & 0 & 52 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6

Find the reduced row echelon form.

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Step 6.1

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

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Step 6.1.1

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

$[\begin{array}{c} - - 1 & - - 1 & - - 1 & - - 1 & - 1 \cdot 1 & - 1 \cdot 1 & - 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 1 & 0 & 0 & 0 & 0 & 0 & 52 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 1 & 0 & 0 & 0 & 0 & 0 & 52 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.2

Perform the row operation $R_{5} = R_{5} - R_{1}$ to make the entry at $5, 1$ a $0$ .

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Step 6.2.1

Perform the row operation $R_{5} = R_{5} - R_{1}$ to make the entry at $5, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 1 - 1 & 0 - 1 & 0 - 1 & 0 - 1 & 0 + 1 & 0 + 1 & 52 - 0 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.2.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & - 1 & - 1 & - 1 & 1 & 1 & 52 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.3

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & - 1 & - 1 & - 1 & 1 & 1 & 52 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.4

Perform the row operation $R_{5} = R_{5} + R_{2}$ to make the entry at $5, 2$ a $0$ .

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Step 6.4.1

Perform the row operation $R_{5} = R_{5} + R_{2}$ to make the entry at $5, 2$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 + 0 & - 1 + 1 \cdot 1 & - 1 + 0 & - 1 + 0 & 1 + 0 & 1 + 0 & 52 + 10 i \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.4.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & - 1 & - 1 & 1 & 1 & 52 + 10 i \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.5

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

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & - 1 & - 1 & 1 & 1 & 52 + 10 i \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.6

Perform the row operation $R_{5} = R_{5} + R_{3}$ to make the entry at $5, 3$ a $0$ .

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Step 6.6.1

Perform the row operation $R_{5} = R_{5} + R_{3}$ to make the entry at $5, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 + 0 & 0 + 0 & - 1 + 1 \cdot 1 & - 1 + 0 & 1 + 0 & 1 + 0 & 52 + 10 i + 1 \cdot 550 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.6.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & - 1 & 1 & 1 & 602 + 10 i \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.7

Swap $R_{5}$ with $R_{4}$ to put a nonzero entry at $4, 4$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & - 1 & 1 & 1 & 602 + 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.8

Multiply each element of $R_{4}$ by $- 1$ to make the entry at $4, 4$ a $1$ .

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Step 6.8.1

Multiply each element of $R_{4}$ by $- 1$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ - 0 & - 0 & - 0 & - - 1 & - 1 \cdot 1 & - 1 \cdot 1 & - (602 + 10 i) \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.8.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & - 32 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.9

Perform the row operation $R_{6} = R_{6} + 32 R_{4}$ to make the entry at $6, 4$ a $0$ .

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Step 6.9.1

Perform the row operation $R_{6} = R_{6} + 32 R_{4}$ to make the entry at $6, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 + 32 \cdot 0 & 0 + 32 \cdot 0 & 0 + 32 \cdot 0 & - 32 + 32 \cdot 1 & 1 + 32 \cdot - 1 & 0 + 32 \cdot - 1 & 0 + 32 (- 602 - 10 i) \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.9.2

Simplify $R_{6}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & 0 & - 31 & - 32 & - 19264 - 320 i \\ 0 & 0 & 0 & 1 & 0 & 0 & 25 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.10

Perform the row operation $R_{7} = R_{7} - R_{4}$ to make the entry at $7, 4$ a $0$ .

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Step 6.10.1

Perform the row operation $R_{7} = R_{7} - R_{4}$ to make the entry at $7, 4$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & 0 & - 31 & - 32 & - 19264 - 320 i \\ 0 - 0 & 0 - 0 & 0 - 0 & 1 - 1 & 0 + 1 & 0 + 1 & 25 - (- 602 - 10 i) \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.10.2

Simplify $R_{7}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & 0 & - 31 & - 32 & - 19264 - 320 i \\ 0 & 0 & 0 & 0 & 1 & 1 & 627 + 10 i \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.11

Swap $R_{6}$ with $R_{5}$ to put a nonzero entry at $5, 5$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & - 31 & - 32 & - 19264 - 320 i \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & 0 & 1 & 1 & 627 + 10 i \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.12

Multiply each element of $R_{5}$ by $- \frac{1}{31}$ to make the entry at $5, 5$ a $1$ .

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Step 6.12.1

Multiply each element of $R_{5}$ by $- \frac{1}{31}$ to make the entry at $5, 5$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ - \frac{1}{31} \cdot 0 & - \frac{1}{31} \cdot 0 & - \frac{1}{31} \cdot 0 & - \frac{1}{31} \cdot 0 & - \frac{1}{31} \cdot - 31 & - \frac{1}{31} \cdot - 32 & - \frac{1}{31} (- 19264 - 320 i) \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & 0 & 1 & 1 & 627 + 10 i \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.12.2

Simplify $R_{5}$ .

Step 6.13

Perform the row operation $R_{7} = R_{7} - R_{5}$ to make the entry at $7, 5$ a $0$ .

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Step 6.13.1

Perform the row operation $R_{7} = R_{7} - R_{5}$ to make the entry at $7, 5$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & \frac{19264}{31} + \frac{320 i}{31} \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 - 0 & 0 - 0 & 0 - 0 & 0 - 0 & 1 - 1 & 1 - \frac{32}{31} & 627 + 10 i - (\frac{19264}{31} + \frac{320 i}{31}) \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.13.2

Simplify $R_{7}$ .

Step 6.14

Perform the row operation $R_{7} = R_{7} + \frac{1}{31} R_{6}$ to make the entry at $7, 6$ a $0$ .

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Step 6.14.1

Perform the row operation $R_{7} = R_{7} + \frac{1}{31} R_{6}$ to make the entry at $7, 6$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & \frac{19264}{31} + \frac{320 i}{31} \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 + \frac{1}{31} \cdot 0 & 0 + \frac{1}{31} \cdot 0 & 0 + \frac{1}{31} \cdot 0 & 0 + \frac{1}{31} \cdot 0 & 0 + \frac{1}{31} \cdot 0 & - \frac{1}{31} + \frac{1}{31} \cdot 1 & \frac{173}{31} - \frac{10 i}{31} + \frac{1}{31} \cdot 51 \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.14.2

Simplify $R_{7}$ .

Step 6.15

Multiply each element of $R_{7}$ by $\frac{1}{\frac{224}{31} - \frac{10 i}{31}}$ to make the entry at $7, 7$ a $1$ .

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Step 6.15.1

Multiply each element of $R_{7}$ by $\frac{1}{\frac{224}{31} - \frac{10 i}{31}}$ to make the entry at $7, 7$ a $1$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & \frac{19264}{31} + \frac{320 i}{31} \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ \frac{0}{\frac{224}{31} - \frac{10 i}{31}} & \frac{0}{\frac{224}{31} - \frac{10 i}{31}} & \frac{0}{\frac{224}{31} - \frac{10 i}{31}} & \frac{0}{\frac{224}{31} - \frac{10 i}{31}} & \frac{0}{\frac{224}{31} - \frac{10 i}{31}} & \frac{0}{\frac{224}{31} - \frac{10 i}{31}} & \frac{\frac{224}{31} - \frac{10 i}{31}}{\frac{224}{31} - \frac{10 i}{31}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 30 - i \end{array}]$

Step 6.15.2

Simplify $R_{7}$ .

Step 6.16

Perform the row operation $R_{8} = R_{8} - (30 - i) R_{7}$ to make the entry at $8, 7$ a $0$ .

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Step 6.16.1

Perform the row operation $R_{8} = R_{8} - (30 - i) R_{7}$ to make the entry at $8, 7$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & \frac{19264}{31} + \frac{320 i}{31} \\ 0 & 0 & 0 & 0 & 0 & 1 & 51 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 - (30 - i) \cdot 0 & 0 - (30 - i) \cdot 0 & 0 - (30 - i) \cdot 0 & 0 - (30 - i) \cdot 0 & 0 - (30 - i) \cdot 0 & 0 - (30 - i) \cdot 0 & 30 - i - (30 - i) \cdot 1 \end{array}]$

Step 6.16.2

Simplify $R_{8}$ .

Step 6.17

Perform the row operation $R_{6} = R_{6} - 51 R_{7}$ to make the entry at $6, 7$ a $0$ .

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Step 6.17.1

Perform the row operation $R_{6} = R_{6} - 51 R_{7}$ to make the entry at $6, 7$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & \frac{19264}{31} + \frac{320 i}{31} \\ 0 - 51 \cdot 0 & 0 - 51 \cdot 0 & 0 - 51 \cdot 0 & 0 - 51 \cdot 0 & 0 - 51 \cdot 0 & 1 - 51 \cdot 0 & 51 - 51 \cdot 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.17.2

Simplify $R_{6}$ .

Step 6.18

Perform the row operation $R_{5} = R_{5} - (\frac{19264}{31} + \frac{320 i}{31}) R_{7}$ to make the entry at $5, 7$ a $0$ .

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Step 6.18.1

Perform the row operation $R_{5} = R_{5} - (\frac{19264}{31} + \frac{320 i}{31}) R_{7}$ to make the entry at $5, 7$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 - (\frac{19264}{31} + \frac{320 i}{31}) \cdot 0 & 0 - (\frac{19264}{31} + \frac{320 i}{31}) \cdot 0 & 0 - (\frac{19264}{31} + \frac{320 i}{31}) \cdot 0 & 0 - (\frac{19264}{31} + \frac{320 i}{31}) \cdot 0 & 1 - (\frac{19264}{31} + \frac{320 i}{31}) \cdot 0 & \frac{32}{31} - (\frac{19264}{31} + \frac{320 i}{31}) \cdot 0 & \frac{19264}{31} + \frac{320 i}{31} - (\frac{19264}{31} + \frac{320 i}{31}) \cdot 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.18.2

Simplify $R_{5}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & - 602 - 10 i \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.19

Perform the row operation $R_{4} = R_{4} - (- 602 - 10 i) R_{7}$ to make the entry at $4, 7$ a $0$ .

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Step 6.19.1

Perform the row operation $R_{4} = R_{4} - (- 602 - 10 i) R_{7}$ to make the entry at $4, 7$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 - (- 602 - 10 i) \cdot 0 & 0 - (- 602 - 10 i) \cdot 0 & 0 - (- 602 - 10 i) \cdot 0 & 1 - (- 602 - 10 i) \cdot 0 & - 1 - (- 602 - 10 i) \cdot 0 & - 1 - (- 602 - 10 i) \cdot 0 & - 602 - 10 i - (- 602 - 10 i) \cdot 1 \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.19.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 550 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.20

Perform the row operation $R_{3} = R_{3} - 550 R_{7}$ to make the entry at $3, 7$ a $0$ .

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Step 6.20.1

Perform the row operation $R_{3} = R_{3} - 550 R_{7}$ to make the entry at $3, 7$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 - 550 \cdot 0 & 0 - 550 \cdot 0 & 1 - 550 \cdot 0 & 0 - 550 \cdot 0 & 0 - 550 \cdot 0 & 0 - 550 \cdot 0 & 550 - 550 \cdot 1 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.20.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 10 i \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.21

Perform the row operation $R_{2} = R_{2} - 10 i R_{7}$ to make the entry at $2, 7$ a $0$ .

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Step 6.21.1

Perform the row operation $R_{2} = R_{2} - 10 i R_{7}$ to make the entry at $2, 7$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 - 10 i \cdot 0 & 1 - 10 i \cdot 0 & 0 - 10 i \cdot 0 & 0 - 10 i \cdot 0 & 0 - 10 i \cdot 0 & 0 - 10 i \cdot 0 & 10 i - 10 i \cdot 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.21.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & \frac{32}{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.22

Perform the row operation $R_{5} = R_{5} - \frac{32}{31} R_{6}$ to make the entry at $5, 6$ a $0$ .

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Step 6.22.1

Perform the row operation $R_{5} = R_{5} - \frac{32}{31} R_{6}$ to make the entry at $5, 6$ a $0$ .

$[\begin{array}{c} 1 & 1 & 1 & 1 & - 1 & - 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 & - 1 & 0 \\ 0 - \frac{32}{31} \cdot 0 & 0 - \frac{32}{31} \cdot 0 & 0 - \frac{32}{31} \cdot 0 & 0 - \frac{32}{31} \cdot 0 & 1 - \frac{32}{31} \cdot 0 & \frac{32}{31} - \frac{32}{31} \cdot 1 & 0 - \frac{32}{31} \cdot 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 6.22.2

Simplify $R_{5}$ .

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

Step 6.23

Perform the row operation $R_{4} = R_{4} + R_{6}$ to make the entry at $4, 6$ a $0$ .

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Step 6.23.1

Perform the row operation $R_{4} = R_{4} + R_{6}$ to make the entry at $4, 6$ a $0$ .

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

Step 6.23.2

Simplify $R_{4}$ .

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

Step 6.24

Perform the row operation $R_{1} = R_{1} + R_{6}$ to make the entry at $1, 6$ a $0$ .

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Step 6.24.1

Perform the row operation $R_{1} = R_{1} + R_{6}$ to make the entry at $1, 6$ a $0$ .

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

Step 6.24.2

Simplify $R_{1}$ .

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

Step 6.25

Perform the row operation $R_{4} = R_{4} + R_{5}$ to make the entry at $4, 5$ a $0$ .

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Step 6.25.1

Perform the row operation $R_{4} = R_{4} + R_{5}$ to make the entry at $4, 5$ a $0$ .

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

Step 6.25.2

Simplify $R_{4}$ .

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

Step 6.26

Perform the row operation $R_{1} = R_{1} + R_{5}$ to make the entry at $1, 5$ a $0$ .

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Step 6.26.1

Perform the row operation $R_{1} = R_{1} + R_{5}$ to make the entry at $1, 5$ a $0$ .

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

Step 6.26.2

Simplify $R_{1}$ .

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

Step 6.27

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

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Step 6.27.1

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

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

Step 6.27.2

Simplify $R_{1}$ .

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

Step 6.28

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

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Step 6.28.1

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

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

Step 6.28.2

Simplify $R_{1}$ .

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

Step 6.29

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

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Step 6.29.1

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

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

Step 6.29.2

Simplify $R_{1}$ .

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

Step 7

Use the result matrix to declare the final solutions to the system of equations.

$a = 0$

$C = 0$

$G = 0$

$I = 0$

$M = 0$

$y = 0$

$0 = 1$

Step 8

Since $0 \neq 1$ , there are no solutions.

No solution