Linear Algebra Examples

$p = 2800$ , $i = 2 %$ , $n = 60$

Step 1

Convert $2 %$ to a decimal.

$p = 2800, i = 0.02, n = 60$

Step 2

Subtract $i$ from both sides of the equation.

$p = 2800, 0 = 0.02 - i, n = 60$

Step 3

Write the system of equations in matrix form.

$[\begin{array}{c} 0 & 1 & 2800 \\ 0 & 0 & 0.02 - i \\ 1 & 0 & 60 \end{array}]$

Step 4

Find the reduced row echelon form.

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Step 4.1

Swap $R_{3}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 1 & 0 & 60 \\ 0 & 0 & 0.02 - i \\ 0 & 1 & 2800 \end{array}]$

Step 4.2

Swap $R_{3}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

$[\begin{array}{c} 1 & 0 & 60 \\ 0 & 1 & 2800 \\ 0 & 0 & 0.02 - i \end{array}]$

Step 4.3

Multiply each element of $R_{3}$ by $\frac{1}{0.02 - i}$ to make the entry at $3, 3$ a $1$ .

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Step 4.3.1

Multiply each element of $R_{3}$ by $\frac{1}{0.02 - i}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & 60 \\ 0 & 1 & 2800 \\ \frac{0}{0.02 - i} & \frac{0}{0.02 - i} & \frac{0.02 - i}{0.02 - i} \end{array}]$

Step 4.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 60 \\ 0 & 1 & 2800 \\ 0 & 0 & 1 \end{array}]$

Step 4.4

Perform the row operation $R_{2} = R_{2} - 2800 R_{3}$ to make the entry at $2, 3$ a $0$ .

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Step 4.4.1

Perform the row operation $R_{2} = R_{2} - 2800 R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 60 \\ 0 - 2800 \cdot 0 & 1 - 2800 \cdot 0 & 2800 - 2800 \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Step 4.4.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 60 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 4.5

Perform the row operation $R_{1} = R_{1} - 60 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 4.5.1

Perform the row operation $R_{1} = R_{1} - 60 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - 60 \cdot 0 & 0 - 60 \cdot 0 & 60 - 60 \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 4.5.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Step 5

Use the result matrix to declare the final solutions to the system of equations.

$n = 0$

$p = 0$

$0 = 1$

Step 6

Since $0 \neq 1$ , there are no solutions.

No solution