Linear Algebra Examples

$[\begin{array}{c} - 9 & 8 \\ 1 & 1 \end{array}] [\begin{array}{c} - 34^{n} & 0 \\ 0 & 34^{n} \end{array}] [\begin{array}{c} - \frac{1}{17} & \frac{8}{17} \\ \frac{1}{17} & \frac{9}{17} \end{array}]$

Step 1

Multiply $[\begin{array}{c} - 9 & 8 \\ 1 & 1 \end{array}] [\begin{array}{c} - 34^{n} & 0 \\ 0 & 34^{n} \end{array}]$ .

Tap for more steps...

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 $2 \times 2$ and the second matrix is $2 \times 2$ .

Step 1.2

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

$[\begin{array}{c} - 9 (- 34^{n}) + 8 \cdot 0 & - 9 \cdot 0 + 8 \cdot 34^{n} \\ 1 (- 34^{n}) + 1 \cdot 0 & 1 \cdot 0 + 1 \cdot 34^{n} \end{array}] [\begin{array}{c} - \frac{1}{17} & \frac{8}{17} \\ \frac{1}{17} & \frac{9}{17} \end{array}]$

Step 1.3

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

$[\begin{array}{c} 9 \cdot 34^{n} & 8 \cdot 34^{n} \\ - 34^{n} & 34^{n} \end{array}] [\begin{array}{c} - \frac{1}{17} & \frac{8}{17} \\ \frac{1}{17} & \frac{9}{17} \end{array}]$

Step 2

Multiply $[\begin{array}{c} 9 \cdot 34^{n} & 8 \cdot 34^{n} \\ - 34^{n} & 34^{n} \end{array}] [\begin{array}{c} - \frac{1}{17} & \frac{8}{17} \\ \frac{1}{17} & \frac{9}{17} \end{array}]$ .

Tap for more steps...

Step 2.1

Step 2.2

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

$[\begin{array}{c} 9 \cdot 34^{n} (- \frac{1}{17}) + 8 \cdot 34^{n} \frac{1}{17} & 9 \cdot 34^{n} \frac{8}{17} + 8 \cdot 34^{n} \frac{9}{17} \\ - 34^{n} (- \frac{1}{17}) + 34^{n} \frac{1}{17} & - 34^{n} \frac{8}{17} + 34^{n} \frac{9}{17} \end{array}]$

Step 2.3

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

$[\begin{array}{c} - \frac{34^{n}}{17} & \frac{144 \cdot 34^{n}}{17} \\ \frac{2 \cdot 34^{n}}{17} & \frac{34^{n}}{17} \end{array}]$

Step 3

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$- \frac{34^{n}}{17} \cdot \frac{34^{n}}{17} - \frac{2 \cdot 34^{n}}{17} \cdot \frac{144 \cdot 34^{n}}{17}$

Step 4

Simplify the determinant.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Multiply $- \frac{34^{n}}{17} \cdot \frac{34^{n}}{17}$ .

Tap for more steps...

Step 4.1.1.1

Multiply $\frac{34^{n}}{17}$ by $\frac{34^{n}}{17}$ .

$- \frac{34^{n} \cdot 34^{n}}{17 \cdot 17} - \frac{2 \cdot 34^{n}}{17} \cdot \frac{144 \cdot 34^{n}}{17}$

Step 4.1.1.2

Multiply $34^{n}$ by $34^{n}$ by adding the exponents.

Tap for more steps...

Step 4.1.1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{34^{n + n}}{17 \cdot 17} - \frac{2 \cdot 34^{n}}{17} \cdot \frac{144 \cdot 34^{n}}{17}$

Step 4.1.1.2.2

Add $n$ and $n$ .

$- \frac{34^{2 n}}{17 \cdot 17} - \frac{2 \cdot 34^{n}}{17} \cdot \frac{144 \cdot 34^{n}}{17}$

Step 4.1.1.3

Multiply $17$ by $17$ .

$- \frac{34^{2 n}}{289} - \frac{2 \cdot 34^{n}}{17} \cdot \frac{144 \cdot 34^{n}}{17}$

Step 4.1.2

Multiply $- \frac{2 \cdot 34^{n}}{17} \cdot \frac{144 \cdot 34^{n}}{17}$ .

Tap for more steps...

Step 4.1.2.1

Multiply $\frac{144 \cdot 34^{n}}{17}$ by $\frac{2 \cdot 34^{n}}{17}$ .

$- \frac{34^{2 n}}{289} - \frac{144 \cdot 34^{n} (2 \cdot 34^{n})}{17 \cdot 17}$

Step 4.1.2.2

Multiply $2$ by $144$ .

$- \frac{34^{2 n}}{289} - \frac{288 \cdot 34^{n} \cdot 34^{n}}{17 \cdot 17}$

Step 4.1.2.3

Multiply $34^{n}$ by $34^{n}$ by adding the exponents.

Tap for more steps...

Step 4.1.2.3.1

Move $34^{n}$ .

$- \frac{34^{2 n}}{289} - \frac{288 \cdot (34^{n} \cdot 34^{n})}{17 \cdot 17}$

Step 4.1.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{34^{2 n}}{289} - \frac{288 \cdot 34^{n + n}}{17 \cdot 17}$

Step 4.1.2.3.3

Add $n$ and $n$ .

$- \frac{34^{2 n}}{289} - \frac{288 \cdot 34^{2 n}}{17 \cdot 17}$

Step 4.1.2.4

Multiply $17$ by $17$ .

$- \frac{34^{2 n}}{289} - \frac{288 \cdot 34^{2 n}}{289}$

Step 4.2

Combine the numerators over the common denominator.

$\frac{- 34^{2 n} - 288 \cdot 34^{2 n}}{289}$

Step 4.3

Subtract $288 \cdot 34^{2 n}$ from $- 34^{2 n}$ .

$\frac{- 289 \cdot 34^{2 n}}{289}$

Step 4.4

Cancel the common factor of $- 289$ and $289$ .

Tap for more steps...

Step 4.4.1

Factor $289$ out of $- 289 \cdot 34^{2 n}$ .

$\frac{289 (- 34^{2 n})}{289}$

Step 4.4.2

Cancel the common factors.

Tap for more steps...

Step 4.4.2.1

Factor $289$ out of $289$ .

$\frac{289 (- 34^{2 n})}{289 (1)}$

Step 4.4.2.2

Cancel the common factor.

$\frac{289 (- 34^{2 n})}{289 \cdot 1}$

Step 4.4.2.3

Rewrite the expression.

$\frac{- 34^{2 n}}{1}$

Step 4.4.2.4

Divide $- 34^{2 n}$ by $1$ .

$- 34^{2 n}$