Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = a ⎡ ⎢ ⎣ \begin{matrix} 1 - 1 8 \end{matrix} ⎤ ⎥ ⎦ + b ⎡ ⎢ ⎣ \begin{matrix} 217 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = a [\begin{array}{c} 1 \\ - 1 \\ 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1

Simplify.

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply

a

$a$ by each element of the matrix.

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a \cdot 1 a \cdot - 1 a \cdot 8 \end{matrix} ⎤ ⎥ ⎦ + b ⎡ ⎢ ⎣ \begin{matrix} 217 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \cdot 1 \\ a \cdot - 1 \\ a \cdot 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.2

Simplify each element in the matrix.

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Step 1.1.2.1

Multiply

a

$a$ by

1

$1$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a a \cdot - 1 a \cdot 8 \end{matrix} ⎤ ⎥ ⎦ + b ⎡ ⎢ ⎣ \begin{matrix} 217 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ a \cdot - 1 \\ a \cdot 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.2.2

Move

- 1

$- 1$ to the left of

a

$a$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - 1 \cdot a a \cdot 8 \end{matrix} ⎤ ⎥ ⎦ + b ⎡ ⎢ ⎣ \begin{matrix} 217 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - 1 \cdot a \\ a \cdot 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.2.3

Rewrite

- 1 a

$- 1 a$ as

- a

$- a$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a a \cdot 8 \end{matrix} ⎤ ⎥ ⎦ + b ⎡ ⎢ ⎣ \begin{matrix} 217 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ a \cdot 8 \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.2.4

Move

8

$8$ to the left of

a

$a$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + b ⎡ ⎢ ⎣ \begin{matrix} 217 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + b [\begin{array}{c} 2 \\ 1 \\ 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + b ⎡ ⎢ ⎣ \begin{matrix} 217 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

Step 1.1.3

Multiply

b

$b$ by each element of the matrix.

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} b \cdot 2 b \cdot 1 b \cdot 7 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} b \cdot 2 \\ b \cdot 1 \\ b \cdot 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.4

Simplify each element in the matrix.

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Step 1.1.4.1

Move

2

$2$ to the left of

b

$b$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b \cdot 1 b \cdot 7 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \cdot 1 \\ b \cdot 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.4.2

Multiply

b

$b$ by

1

$1$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b b \cdot 7 \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ b \cdot 7 \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.4.3

Move

7

$7$ to the left of

b

$b$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + c [\begin{array}{c} 1 \\ 3 \\ - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + c ⎡ ⎢ ⎣ \begin{matrix} 13 - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

Step 1.1.5

Multiply

c

$c$ by each element of the matrix.

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c \cdot 1 c \cdot 3 c \cdot - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} c \cdot 1 \\ c \cdot 3 \\ c \cdot - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.6

Simplify each element in the matrix.

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Step 1.1.6.1

Multiply

c

$c$ by

1

$1$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c c \cdot 3 c \cdot - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} c \\ c \cdot 3 \\ c \cdot - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.6.2

Move

3

$3$ to the left of

c

$c$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c c \cdot - 4 \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} c \\ 3 c \\ c \cdot - 4 \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

Step 1.1.6.3

Move

- 4

$- 4$ to the left of

c

$c$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a \\ - a \\ 8 a \end{array}] + [\begin{array}{c} 2 b \\ b \\ 7 b \end{array}] + [\begin{array}{c} c \\ 3 c \\ - 4 c \end{array}] + d [\begin{array}{c} 3 \\ - 4 \\ 36 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + d ⎡ ⎢ ⎣ \begin{matrix} 3 - 4 36 \end{matrix} ⎤ ⎥ ⎦

Step 1.1.7

Multiply

d

$d$ by each element of the matrix.

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} d \cdot 3 d \cdot - 4 d \cdot 36 \end{matrix} ⎤ ⎥ ⎦

Step 1.1.8

Simplify each element in the matrix.

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Step 1.1.8.1

Move

3

$3$ to the left of

d

$d$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 3 d d \cdot - 4 d \cdot 36 \end{matrix} ⎤ ⎥ ⎦

Step 1.1.8.2

Move

- 4

$- 4$ to the left of

d

$d$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 3 d - 4 d d \cdot 36 \end{matrix} ⎤ ⎥ ⎦

Step 1.1.8.3

Move

36

$36$ to the left of

d

$d$ .

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 3 d - 4 d 36 d \end{matrix} ⎤ ⎥ ⎦

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 3 d - 4 d 36 d \end{matrix} ⎤ ⎥ ⎦

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a - a 8 a \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 2 b b 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 3 d - 4 d 36 d \end{matrix} ⎤ ⎥ ⎦

Step 1.2

Add the corresponding elements.

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a + 2 b - a + b 8 a + 7 b \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} c 3 c - 4 c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 3 d - 4 d 36 d \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a + 2 b \\ - a + b \\ 8 a + 7 b \end{array}] + [\begin{array}{c} c \\ 3 c \\ - 4 c \end{array}] + [\begin{array}{c} 3 d \\ - 4 d \\ 36 d \end{array}]$

Step 1.3

Add the corresponding elements.

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a + 2 b + c - a + b + 3 c 8 a + 7 b - 4 c \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} 3 d - 4 d 36 d \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a + 2 b + c \\ - a + b + 3 c \\ 8 a + 7 b - 4 c \end{array}] + [\begin{array}{c} 3 d \\ - 4 d \\ 36 d \end{array}]$

Step 1.4

Add the corresponding elements.

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a + 2 b + c + 3 d - a + b + 3 c - 4 d 8 a + 7 b - 4 c + 36 d \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a + 2 b + c + 3 d \\ - a + b + 3 c - 4 d \\ 8 a + 7 b - 4 c + 36 d \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 10 - 15 131 \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} a + 2 b + c + 3 d - a + b + 3 c - 4 d 8 a + 7 b - 4 c + 36 d \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 10 \\ - 15 \\ 131 \end{array}] = [\begin{array}{c} a + 2 b + c + 3 d \\ - a + b + 3 c - 4 d \\ 8 a + 7 b - 4 c + 36 d \end{array}]$

Step 2

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

10 = a + 2 b + c + 3 d

$10 = a + 2 b + c + 3 d$

- 15 = - a + b + 3 c - 4 d

$- 15 = - a + b + 3 c - 4 d$

131 = 8 a + 7 b - 4 c + 36 d

$131 = 8 a + 7 b - 4 c + 36 d$