Linear Algebra Examples

a ⎡ ⎢ ⎣ \begin{matrix} 12 - 1 \end{matrix} ⎤ ⎥ ⎦ + b ⎡ ⎢ ⎣ \begin{matrix} 13 - 2 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 37 - 4 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 000 \end{matrix} ⎤ ⎥ ⎦

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

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.

$a = - 2 c$

$b = - c$