Linear Algebra Examples

$[\begin{array}{c} 3 & - 4 & 0 \\ 2 & 8 & - 1 \end{array}] + B = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$

Step 1

Find the function rule.

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Step 1.1

Check if the function rule is linear.

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Step 1.1.1

To find if the table follows a function rule, check to see if the values follow the linear form $y = a x + b$ .

$y = a x + b$

Step 1.1.2

Build a set of equations from the table such that $y = a x + b$ .

$0 = a (3) + b 0 = a (- 4) + b 0 = a (0) + b 0 = a (2) + b 0 = a (8) + b 0 = a (- 1) + b$

Step 1.1.3

Calculate the values of $a$ and $b$ .

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Step 1.1.3.1

Rewrite the equation as $b = 0$ .

$b = 0$

$0 = a (3) + b$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2

Replace all occurrences of $b$ with $0$ in each equation.

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Step 1.1.3.2.1

Replace all occurrences of $b$ in $0 = a (3) + b$ with $0$ .

$0 = a (3) + 0$

$b = 0$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.2

Simplify $0 = a (3) + 0$ .

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Step 1.1.3.2.2.1

Simplify the left side.

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Step 1.1.3.2.2.1.1

Remove parentheses.

$0 = a (3) + 0$

$b = 0$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = a (3) + 0$

$b = 0$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.2.2

Simplify the right side.

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Step 1.1.3.2.2.2.1

Simplify $a (3) + 0$ .

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Step 1.1.3.2.2.2.1.1

Add $a (3)$ and $0$ .

$0 = a (3)$

$b = 0$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.2.2.1.2

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

$0 = 3 a$

$b = 0$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = 3 a$

$b = 0$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = 3 a$

$b = 0$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = 3 a$

$b = 0$

$0 = a (- 4) + b$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.3

Replace all occurrences of $b$ in $0 = a (- 4) + b$ with $0$ .

$0 = a (- 4) + 0$

$0 = 3 a$

$b = 0$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.4

Simplify $0 = a (- 4) + 0$ .

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Step 1.1.3.2.4.1

Simplify the left side.

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Step 1.1.3.2.4.1.1

Remove parentheses.

$0 = a (- 4) + 0$

$0 = 3 a$

$b = 0$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = a (- 4) + 0$

$0 = 3 a$

$b = 0$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.4.2

Simplify the right side.

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Step 1.1.3.2.4.2.1

Simplify $a (- 4) + 0$ .

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Step 1.1.3.2.4.2.1.1

Add $a (- 4)$ and $0$ .

$0 = a (- 4)$

$0 = 3 a$

$b = 0$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.4.2.1.2

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

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (2) + b$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.5

Replace all occurrences of $b$ in $0 = a (2) + b$ with $0$ .

$0 = a (2) + 0$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.6

Simplify $0 = a (2) + 0$ .

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Step 1.1.3.2.6.1

Simplify the left side.

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Step 1.1.3.2.6.1.1

Remove parentheses.

$0 = a (2) + 0$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = a (2) + 0$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.6.2

Simplify the right side.

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Step 1.1.3.2.6.2.1

Simplify $a (2) + 0$ .

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Step 1.1.3.2.6.2.1.1

Add $a (2)$ and $0$ .

$0 = a (2)$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.6.2.1.2

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

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (8) + b$

$0 = a (- 1) + b$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (8) + b$

$0 = a (- 1) + b$

Step 1.1.3.2.7

Replace all occurrences of $b$ in $0 = a (8) + b$ with $0$ .

$0 = a (8) + 0$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + b$

Step 1.1.3.2.8

Simplify $0 = a (8) + 0$ .

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Step 1.1.3.2.8.1

Simplify the left side.

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Step 1.1.3.2.8.1.1

Remove parentheses.

$0 = a (8) + 0$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + b$

$0 = a (8) + 0$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + b$

Step 1.1.3.2.8.2

Simplify the right side.

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Step 1.1.3.2.8.2.1

Simplify $a (8) + 0$ .

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Step 1.1.3.2.8.2.1.1

Add $a (8)$ and $0$ .

$0 = a (8)$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + b$

Step 1.1.3.2.8.2.1.2

Move $8$ to the left of $a$ .

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + b$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + b$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + b$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + b$

Step 1.1.3.2.9

Replace all occurrences of $b$ in $0 = a (- 1) + b$ with $0$ .

$0 = a (- 1) + 0$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.2.10

Simplify $0 = a (- 1) + 0$ .

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Step 1.1.3.2.10.1

Simplify the left side.

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Step 1.1.3.2.10.1.1

Remove parentheses.

$0 = a (- 1) + 0$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = a (- 1) + 0$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.2.10.2

Simplify the right side.

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Step 1.1.3.2.10.2.1

Simplify $a (- 1) + 0$ .

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Step 1.1.3.2.10.2.1.1

Add $a (- 1)$ and $0$ .

$0 = a (- 1)$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.2.10.2.1.2

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

$0 = - 1 a$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.2.10.2.1.3

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

$0 = - a$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = - a$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = - a$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = - a$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = - a$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.3

Solve for $a$ in $0 = - a$ .

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Step 1.1.3.3.1

Rewrite the equation as $- a = 0$ .

$- a = 0$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.3.2

Divide each term in $- a = 0$ by $- 1$ and simplify.

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Step 1.1.3.3.2.1

Divide each term in $- a = 0$ by $- 1$ .

$\frac{- a}{- 1} = \frac{0}{- 1}$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.3.2.2

Simplify the left side.

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Step 1.1.3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{a}{1} = \frac{0}{- 1}$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.3.2.2.2

Divide $a$ by $1$ .

$a = \frac{0}{- 1}$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$a = \frac{0}{- 1}$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.3.2.3

Simplify the right side.

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Step 1.1.3.3.2.3.1

Divide $0$ by $- 1$ .

$a = 0$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$a = 0$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$a = 0$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$a = 0$

$0 = 8 a$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.4

Replace all occurrences of $a$ with $0$ in each equation.

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Step 1.1.3.4.1

Replace all occurrences of $a$ in $0 = 8 a$ with $0$ .

$0 = 8 (0)$

$a = 0$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.4.2

Simplify the right side.

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Step 1.1.3.4.2.1

Multiply $8$ by $0$ .

$0 = 0$

$a = 0$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = 0$

$a = 0$

$0 = 2 a$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.4.3

Replace all occurrences of $a$ in $0 = 2 a$ with $0$ .

$0 = 2 (0)$

$0 = 0$

$a = 0$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.4.4

Simplify the right side.

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Step 1.1.3.4.4.1

Multiply $2$ by $0$ .

$0 = 0$

$a = 0$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

$0 = 0$

$a = 0$

$0 = - 4 a$

$0 = 3 a$

$b = 0$

Step 1.1.3.4.5

Replace all occurrences of $a$ in $0 = - 4 a$ with $0$ .

$0 = - 4 \cdot 0$

$0 = 0$

$a = 0$

$0 = 3 a$

$b = 0$

Step 1.1.3.4.6

Simplify the right side.

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Step 1.1.3.4.6.1

Multiply $- 4$ by $0$ .

$0 = 0$

$a = 0$

$0 = 3 a$

$b = 0$

$0 = 0$

$a = 0$

$0 = 3 a$

$b = 0$

Step 1.1.3.4.7

Replace all occurrences of $a$ in $0 = 3 a$ with $0$ .

$0 = 3 (0)$

$0 = 0$

$a = 0$

$b = 0$

Step 1.1.3.4.8

Simplify the right side.

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Step 1.1.3.4.8.1

Multiply $3$ by $0$ .

$0 = 0$

$a = 0$

$b = 0$

$0 = 0$

$a = 0$

$b = 0$

$0 = 0$

$a = 0$

$b = 0$

Step 1.1.3.5

Remove any equations from the system that are always true.

$a = 0$

$b = 0$

Step 1.1.3.6

List all of the solutions.

$a = 0, b = 0$

Step 1.1.4

Calculate the value of $y$ using each $x$ value in the relation and compare this value to the given $y$ value in the relation.

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Step 1.1.4.1

Calculate the value of $y$ when $a = 0$ , $b = 0$ , and $x = 3$ .

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Step 1.1.4.1.1

Multiply $0$ by $3$ .

$y = 0 + 0$

Step 1.1.4.1.2

Add $0$ and $0$ .

$y = 0$

Step 1.1.4.2

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 3$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Step 1.1.4.3

Calculate the value of $y$ when $a = 0$ , $b = 0$ , and $x = - 4$ .

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Step 1.1.4.3.1

Multiply $0$ by $- 4$ .

$y = 0 + 0$

Step 1.1.4.3.2

Add $0$ and $0$ .

$y = 0$

Step 1.1.4.4

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 4$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Step 1.1.4.5

Calculate the value of $y$ when $a = 0$ , $b = 0$ , and $x = 0$ .

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Step 1.1.4.5.1

Multiply $0$ by $0$ .

$y = 0 + 0$

Step 1.1.4.5.2

Add $0$ and $0$ .

$y = 0$

Step 1.1.4.6

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 0$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Step 1.1.4.7

Calculate the value of $y$ when $a = 0$ , $b = 0$ , and $x = 2$ .

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Step 1.1.4.7.1

Multiply $0$ by $2$ .

$y = 0 + 0$

Step 1.1.4.7.2

Add $0$ and $0$ .

$y = 0$

Step 1.1.4.8

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 2$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Step 1.1.4.9

Calculate the value of $y$ when $a = 0$ , $b = 0$ , and $x = 8$ .

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Step 1.1.4.9.1

Multiply $0$ by $8$ .

$y = 0 + 0$

Step 1.1.4.9.2

Add $0$ and $0$ .

$y = 0$

Step 1.1.4.10

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = 8$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Step 1.1.4.11

Calculate the value of $y$ when $a = 0$ , $b = 0$ , and $x = - 1$ .

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Step 1.1.4.11.1

Multiply $0$ by $- 1$ .

$y = 0 + 0$

Step 1.1.4.11.2

Add $0$ and $0$ .

$y = 0$

Step 1.1.4.12

If the table has a linear function rule, $y = y$ for the corresponding $x$ value, $x = - 1$ . This check passes since $y = 0$ and $y = 0$ .

$0 = 0$

Step 1.1.4.13

Since $y = y$ for the corresponding $x$ values, the function is linear.

The function is linear

Step 1.2

Since all $y = y$ , the function is linear and follows the form $y = 0$ .

$y = 0$

Step 2

List all of the solutions.

Step 3

Nothing further can be done with this topic. Please check the expression entered or try another topic.

$[\begin{array}{c} 3 & - 4 & 0 \\ 2 & 8 & - 1 \end{array}] + B = [\begin{array}{c} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$