Linear Algebra Examples

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8}{3} \frac{8}{3} \frac{8}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} - \frac{19}{14} \frac{57}{28} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8}{3} \\ \frac{8}{3} \\ \frac{8}{3} \end{array}] + [\begin{array}{c} - \frac{19}{14} \\ \frac{57}{28} \\ - \frac{19}{28} \end{array}]$

Step 1

Add the corresponding elements.

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8}{3} - \frac{19}{14} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8}{3} - \frac{19}{14} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2

Simplify each element.

Tap for more steps...

Step 2.1

To write

\frac{8}{3}

$\frac{8}{3}$ as a fraction with a common denominator, multiply by

\frac{14}{14}

$\frac{14}{14}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8}{3} \cdot \frac{14}{14} - \frac{19}{14} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8}{3} \cdot \frac{14}{14} - \frac{19}{14} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.2

To write

- \frac{19}{14}

$- \frac{19}{14}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8}{3} \cdot \frac{14}{14} - \frac{19}{14} \cdot \frac{3}{3} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8}{3} \cdot \frac{14}{14} - \frac{19}{14} \cdot \frac{3}{3} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.3

Write each expression with a common denominator of

42

$42$ , by multiplying each by an appropriate factor of

1

$1$ .

Tap for more steps...

Step 2.3.1

Multiply

\frac{8}{3}

$\frac{8}{3}$ by

\frac{14}{14}

$\frac{14}{14}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8 \cdot 14}{3 \cdot 14} - \frac{19}{14} \cdot \frac{3}{3} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8 \cdot 14}{3 \cdot 14} - \frac{19}{14} \cdot \frac{3}{3} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.3.2

Multiply

3

$3$ by

14

$14$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8 \cdot 14}{42} - \frac{19}{14} \cdot \frac{3}{3} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8 \cdot 14}{42} - \frac{19}{14} \cdot \frac{3}{3} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.3.3

Multiply

\frac{19}{14}

$\frac{19}{14}$ by

\frac{3}{3}

$\frac{3}{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8 \cdot 14}{42} - \frac{19 \cdot 3}{14 \cdot 3} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8 \cdot 14}{42} - \frac{19 \cdot 3}{14 \cdot 3} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.3.4

Multiply

14

$14$ by

3

$3$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8 \cdot 14}{42} - \frac{19 \cdot 3}{42} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8 \cdot 14}{42} - \frac{19 \cdot 3}{42} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8 \cdot 14}{42} - \frac{19 \cdot 3}{42} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8 \cdot 14}{42} - \frac{19 \cdot 3}{42} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.4

Combine the numerators over the common denominator.

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{8 \cdot 14 - 19 \cdot 3}{42} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{8 \cdot 14 - 19 \cdot 3}{42} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.5

Simplify the numerator.

Tap for more steps...

Step 2.5.1

Multiply

8

$8$ by

14

$14$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{112 - 19 \cdot 3}{42} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{112 - 19 \cdot 3}{42} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.5.2

Multiply

- 19

$- 19$ by

3

$3$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{112 - 57}{42} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{112 - 57}{42} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.5.3

Subtract

57

$57$ from

112

$112$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{55}{42} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{55}{42} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{55}{42} \frac{8}{3} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{55}{42} \\ \frac{8}{3} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.6

To write

\frac{8}{3}

$\frac{8}{3}$ as a fraction with a common denominator, multiply by

\frac{28}{28}

$\frac{28}{28}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{55}{42} \frac{8}{3} \cdot \frac{28}{28} + \frac{57}{28} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{55}{42} \\ \frac{8}{3} \cdot \frac{28}{28} + \frac{57}{28} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.7

To write

\frac{57}{28}

$\frac{57}{28}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{55}{42} \frac{8}{3} \cdot \frac{28}{28} + \frac{57}{28} \cdot \frac{3}{3} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{55}{42} \\ \frac{8}{3} \cdot \frac{28}{28} + \frac{57}{28} \cdot \frac{3}{3} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.8

Write each expression with a common denominator of

84

$84$ , by multiplying each by an appropriate factor of

1

$1$ .

Tap for more steps...

Step 2.8.1

Multiply

\frac{8}{3}

$\frac{8}{3}$ by

\frac{28}{28}

$\frac{28}{28}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{55}{42} \frac{8 \cdot 28}{3 \cdot 28} + \frac{57}{28} \cdot \frac{3}{3} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{55}{42} \\ \frac{8 \cdot 28}{3 \cdot 28} + \frac{57}{28} \cdot \frac{3}{3} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.8.2

Multiply

3

$3$ by

28

$28$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{55}{42} \frac{8 \cdot 28}{84} + \frac{57}{28} \cdot \frac{3}{3} \frac{8}{3} - \frac{19}{28} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{55}{42} \\ \frac{8 \cdot 28}{84} + \frac{57}{28} \cdot \frac{3}{3} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.8.3

Multiply

$\frac{57}{28}$ by

$\frac{3}{3}$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{8 \cdot 28}{84} + \frac{57 \cdot 3}{28 \cdot 3} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.8.4

Multiply

$28$ by

$3$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{8 \cdot 28}{84} + \frac{57 \cdot 3}{84} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.9

Combine the numerators over the common denominator.

$[\begin{array}{c} \frac{55}{42} \\ \frac{8 \cdot 28 + 57 \cdot 3}{84} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.10

Simplify the numerator.

Tap for more steps...

Step 2.10.1

Multiply

$8$ by

$28$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{224 + 57 \cdot 3}{84} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.10.2

Multiply

$57$ by

$3$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{224 + 171}{84} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.10.3

Add

$224$ and

$171$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{8}{3} - \frac{19}{28} \end{array}]$

Step 2.11

To write

$\frac{8}{3}$ as a fraction with a common denominator, multiply by

$\frac{28}{28}$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{8}{3} \cdot \frac{28}{28} - \frac{19}{28} \end{array}]$

Step 2.12

To write

$- \frac{19}{28}$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{8}{3} \cdot \frac{28}{28} - \frac{19}{28} \cdot \frac{3}{3} \end{array}]$

Step 2.13

Write each expression with a common denominator of

$84$ , by multiplying each by an appropriate factor of

$1$ .

Tap for more steps...

Step 2.13.1

Multiply

$\frac{8}{3}$ by

$\frac{28}{28}$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{8 \cdot 28}{3 \cdot 28} - \frac{19}{28} \cdot \frac{3}{3} \end{array}]$

Step 2.13.2

Multiply

$3$ by

$28$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{8 \cdot 28}{84} - \frac{19}{28} \cdot \frac{3}{3} \end{array}]$

Step 2.13.3

Multiply

$\frac{19}{28}$ by

$\frac{3}{3}$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{8 \cdot 28}{84} - \frac{19 \cdot 3}{28 \cdot 3} \end{array}]$

Step 2.13.4

Multiply

$28$ by

$3$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{8 \cdot 28}{84} - \frac{19 \cdot 3}{84} \end{array}]$

Step 2.14

Combine the numerators over the common denominator.

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{8 \cdot 28 - 19 \cdot 3}{84} \end{array}]$

Step 2.15

Simplify the numerator.

Tap for more steps...

Step 2.15.1

Multiply

$8$ by

$28$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{224 - 19 \cdot 3}{84} \end{array}]$

Step 2.15.2

Multiply

$- 19$ by

$3$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{224 - 57}{84} \end{array}]$

Step 2.15.3

Subtract

$57$ from

$224$ .

$[\begin{array}{c} \frac{55}{42} \\ \frac{395}{84} \\ \frac{167}{84} \end{array}]$