Linear Algebra Examples

$[\begin{array}{c} 0 & 3 & 4 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{array}] [\begin{array}{c} a \\ b \\ d \end{array}] = [\begin{array}{c} 1 \\ 0 \\ 0 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 0 & 3 & 4 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{array}] [\begin{array}{c} a \\ b \\ d \end{array}]$ .

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Step 1.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $3 \times 3$ and the second matrix is $3 \times 1$ .

Step 1.2

Multiply each row in the first matrix by each column in the second matrix.

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

Step 1.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 3 b + 4 d \\ 3 d \\ 0 \end{array}] = [\begin{array}{c} 1 \\ 0 \\ 0 \end{array}]$

Step 2

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

$3 b + 4 d = 1$

$3 d = 0$

$0 = 0$

Step 3

Divide each term in $3 d = 0$ by $3$ and simplify.

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Step 3.1

Divide each term in $3 d = 0$ by $3$ .

$\frac{3 d}{3} = \frac{0}{3}$

$3 b + 4 d = 1$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 d}{3} = \frac{0}{3}$

$3 b + 4 d = 1$

Step 3.2.1.2

Divide $d$ by $1$ .

$d = \frac{0}{3}$

$3 b + 4 d = 1$

$d = \frac{0}{3}$

$3 b + 4 d = 1$

$d = \frac{0}{3}$

$3 b + 4 d = 1$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $3$ .

$d = 0$

$3 b + 4 d = 1$

$d = 0$

$3 b + 4 d = 1$

$d = 0$

$3 b + 4 d = 1$

Step 4

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

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Step 4.1

Replace all occurrences of $d$ in $3 b + 4 d = 1$ with $0$ .

$3 b + 4 (0) = 1$

$d = 0$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify $3 b + 4 (0)$ .

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Step 4.2.1.1

Multiply $4$ by $0$ .

$3 b + 0 = 1$

$d = 0$

Step 4.2.1.2

Add $3 b$ and $0$ .

$3 b = 1$

$d = 0$

$3 b = 1$

$d = 0$

$3 b = 1$

$d = 0$

$3 b = 1$

$d = 0$

Step 5

Divide each term in $3 b = 1$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 b = 1$ by $3$ .

$\frac{3 b}{3} = \frac{1}{3}$

$d = 0$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 b}{3} = \frac{1}{3}$

$d = 0$

Step 5.2.1.2

Divide $b$ by $1$ .

$b = \frac{1}{3}$

$d = 0$

$b = \frac{1}{3}$

$d = 0$

$b = \frac{1}{3}$

$d = 0$

$b = \frac{1}{3}$

$d = 0$

Step 6

Solve the system of equations.

$b = \frac{1}{3} d = 0$

Step 7

List all of the solutions.

$b = \frac{1}{3}, d = 0$