Linear Algebra Examples

$a [\begin{array}{c} 1 \\ 2 \\ - 1 \end{array}] + b [\begin{array}{c} 1 \\ 3 \\ - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1

Simplify.

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $a$ by each element of the matrix.

$[\begin{array}{c} a \cdot 1 \\ a \cdot 2 \\ a \cdot - 1 \end{array}] + b [\begin{array}{c} 1 \\ 3 \\ - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.2

Simplify each element in the matrix.

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Step 1.1.2.1

Multiply $a$ by $1$ .

$[\begin{array}{c} a \\ a \cdot 2 \\ a \cdot - 1 \end{array}] + b [\begin{array}{c} 1 \\ 3 \\ - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.2.2

Move $2$ to the left of $a$ .

$[\begin{array}{c} a \\ 2 a \\ a \cdot - 1 \end{array}] + b [\begin{array}{c} 1 \\ 3 \\ - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.2.3

Move $- 1$ to the left of $a$ .

$[\begin{array}{c} a \\ 2 a \\ - 1 \cdot a \end{array}] + b [\begin{array}{c} 1 \\ 3 \\ - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.2.4

Rewrite $- 1 a$ as $- a$ .

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + b [\begin{array}{c} 1 \\ 3 \\ - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.3

Multiply $b$ by each element of the matrix.

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + [\begin{array}{c} b \cdot 1 \\ b \cdot 3 \\ b \cdot - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.4

Simplify each element in the matrix.

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Step 1.1.4.1

Multiply $b$ by $1$ .

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + [\begin{array}{c} b \\ b \cdot 3 \\ b \cdot - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.4.2

Move $3$ to the left of $b$ .

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + [\begin{array}{c} b \\ 3 b \\ b \cdot - 2 \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.4.3

Move $- 2$ to the left of $b$ .

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + [\begin{array}{c} b \\ 3 b \\ - 2 b \end{array}] + c [\begin{array}{c} 3 \\ 7 \\ - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.5

Multiply $c$ by each element of the matrix.

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + [\begin{array}{c} b \\ 3 b \\ - 2 b \end{array}] + [\begin{array}{c} c \cdot 3 \\ c \cdot 7 \\ c \cdot - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.6

Simplify each element in the matrix.

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Step 1.1.6.1

Move $3$ to the left of $c$ .

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + [\begin{array}{c} b \\ 3 b \\ - 2 b \end{array}] + [\begin{array}{c} 3 c \\ c \cdot 7 \\ c \cdot - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.6.2

Move $7$ to the left of $c$ .

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + [\begin{array}{c} b \\ 3 b \\ - 2 b \end{array}] + [\begin{array}{c} 3 c \\ 7 c \\ c \cdot - 4 \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.1.6.3

Move $- 4$ to the left of $c$ .

$[\begin{array}{c} a \\ 2 a \\ - a \end{array}] + [\begin{array}{c} b \\ 3 b \\ - 2 b \end{array}] + [\begin{array}{c} 3 c \\ 7 c \\ - 4 c \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.2

Add the corresponding elements.

$[\begin{array}{c} a + b \\ 2 a + 3 b \\ - a - 2 b \end{array}] + [\begin{array}{c} 3 c \\ 7 c \\ - 4 c \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 1.3

Add the corresponding elements.

$[\begin{array}{c} a + b + 3 c \\ 2 a + 3 b + 7 c \\ - a - 2 b - 4 c \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$a + b + 3 c = 0$

$2 a + 3 b + 7 c = 0$

$- a - 2 b - 4 c = 0$

Step 3

Move all terms not containing $a$ to the right side of the equation.

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Step 3.1

Subtract $b$ from both sides of the equation.

$a + 3 c = - b$

$2 a + 3 b + 7 c = 0$

$- a - 2 b - 4 c = 0$

Step 3.2

Subtract $3 c$ from both sides of the equation.

$a = - b - 3 c$

$2 a + 3 b + 7 c = 0$

$- a - 2 b - 4 c = 0$

$a = - b - 3 c$

$2 a + 3 b + 7 c = 0$

$- a - 2 b - 4 c = 0$

Step 4

Replace all occurrences of $a$ with $- b - 3 c$ in each equation.

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Step 4.1

Replace all occurrences of $a$ in $2 a + 3 b + 7 c = 0$ with $- b - 3 c$ .

$2 (- b - 3 c) + 3 b + 7 c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $2 (- b - 3 c) + 3 b + 7 c$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the distributive property.

$2 (- b) + 2 (- 3 c) + 3 b + 7 c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

Step 4.2.1.1.2

Multiply $- 1$ by $2$ .

$- 2 b + 2 (- 3 c) + 3 b + 7 c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

Step 4.2.1.1.3

Multiply $- 3$ by $2$ .

$- 2 b - 6 c + 3 b + 7 c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

$- 2 b - 6 c + 3 b + 7 c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

Step 4.2.1.2

Simplify by adding terms.

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Step 4.2.1.2.1

Add $- 2 b$ and $3 b$ .

$b - 6 c + 7 c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

Step 4.2.1.2.2

Add $- 6 c$ and $7 c$ .

$b + c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

$- a - 2 b - 4 c = 0$

Step 4.3

Replace all occurrences of $a$ in $- a - 2 b - 4 c = 0$ with $- b - 3 c$ .

$- (- b - 3 c) - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $- (- b - 3 c) - 2 b - 4 c$ .

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Step 4.4.1.1

Simplify each term.

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Step 4.4.1.1.1

Apply the distributive property.

$b - (- 3 c) - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

Step 4.4.1.1.2

Multiply $- - b$ .

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Step 4.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 b - (- 3 c) - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

Step 4.4.1.1.2.2

Multiply $b$ by $1$ .

$b - (- 3 c) - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

$b - (- 3 c) - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

Step 4.4.1.1.3

Multiply $- 3$ by $- 1$ .

$b + 3 c - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

$b + 3 c - 2 b - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

Step 4.4.1.2

Simplify by adding terms.

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Step 4.4.1.2.1

Subtract $2 b$ from $b$ .

$- b + 3 c - 4 c = 0$

$b + c = 0$

$a = - b - 3 c$

Step 4.4.1.2.2

Subtract $4 c$ from $3 c$ .

$- b - c = 0$

$b + c = 0$

$a = - b - 3 c$

$- b - c = 0$

$b + c = 0$

$a = - b - 3 c$

$- b - c = 0$

$b + c = 0$

$a = - b - 3 c$

$- b - c = 0$

$b + c = 0$

$a = - b - 3 c$

$- b - c = 0$

$b + c = 0$

$a = - b - 3 c$

Step 5

Subtract $c$ from both sides of the equation.

$b = - c$

$- b - c = 0$

$a = - b - 3 c$

Step 6

Replace all occurrences of $b$ with $- c$ in each equation.

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Step 6.1

Replace all occurrences of $b$ in $- b - c = 0$ with $- c$ .

$- (- c) - c = 0$

$b = - c$

$a = - b - 3 c$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify $- (- c) - c$ .

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Step 6.2.1.1

Multiply $- (- c)$ .

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Step 6.2.1.1.1

Multiply $- 1$ by $- 1$ .

$1 c - c = 0$

$b = - c$

$a = - b - 3 c$

Step 6.2.1.1.2

Multiply $c$ by $1$ .

$c - c = 0$

$b = - c$

$a = - b - 3 c$

$c - c = 0$

$b = - c$

$a = - b - 3 c$

Step 6.2.1.2

Subtract $c$ from $c$ .

$0 = 0$

$b = - c$

$a = - b - 3 c$

$0 = 0$

$b = - c$

$a = - b - 3 c$

$0 = 0$

$b = - c$

$a = - b - 3 c$

Step 6.3

Replace all occurrences of $b$ in $a = - b - 3 c$ with $- c$ .

$a = - (- c) - 3 c$

$0 = 0$

$b = - c$

Step 6.4

Simplify the right side.

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Step 6.4.1

Simplify $- (- c) - 3 c$ .

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Step 6.4.1.1

Multiply $- (- c)$ .

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Step 6.4.1.1.1

Multiply $- 1$ by $- 1$ .

$a = 1 c - 3 c$

$0 = 0$

$b = - c$

Step 6.4.1.1.2

Multiply $c$ by $1$ .

$a = c - 3 c$

$0 = 0$

$b = - c$

$a = c - 3 c$

$0 = 0$

$b = - c$

Step 6.4.1.2

Subtract $3 c$ from $c$ .

$a = - 2 c$

$0 = 0$

$b = - c$

$a = - 2 c$

$0 = 0$

$b = - c$

$a = - 2 c$

$0 = 0$

$b = - c$

$a = - 2 c$

$0 = 0$

$b = - c$

Step 7

Remove any equations from the system that are always true.

$a = - 2 c$

$b = - c$