Finite Math Examples

$[\begin{array}{c} 1.35 \\ 3.2 \\ 1.4 \end{array}] [\begin{array}{c} 1200 & 500 & 5500 \end{array}]$

Step 1

Multiply $[\begin{array}{c} 1.35 \\ 3.2 \\ 1.4 \end{array}] [\begin{array}{c} 1200 & 500 & 5500 \end{array}]$ .

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 $3 \times 1$ and the second matrix is $1 \times 3$ .

Step 1.2

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

$[\begin{array}{c} 1.35 \cdot 1200 & 1.35 \cdot 500 & 1.35 \cdot 5500 \\ 3.2 \cdot 1200 & 3.2 \cdot 500 & 3.2 \cdot 5500 \\ 1.4 \cdot 1200 & 1.4 \cdot 500 & 1.4 \cdot 5500 \end{array}]$

Step 1.3

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

$[\begin{array}{c} 1620 & 675 & 7425 \\ 3840 & 1600 & 17600 \\ 1680 & 700 & 7700 \end{array}]$

Step 2

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

Tap for more steps...

Step 2.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} 1600 & 17600 \\ 700 & 7700 \end{array} |$

Step 2.4

Multiply element $a_{11}$ by its cofactor.

$1620 | \begin{array}{c} 1600 & 17600 \\ 700 & 7700 \end{array} |$

Step 2.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 3840 & 17600 \\ 1680 & 7700 \end{array} |$

Step 2.6

Multiply element $a_{12}$ by its cofactor.

$- 675 | \begin{array}{c} 3840 & 17600 \\ 1680 & 7700 \end{array} |$

Step 2.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 2.8

Multiply element $a_{13}$ by its cofactor.

$7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 2.9

Add the terms together.

$1620 | \begin{array}{c} 1600 & 17600 \\ 700 & 7700 \end{array} | - 675 | \begin{array}{c} 3840 & 17600 \\ 1680 & 7700 \end{array} | + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 3

Evaluate $| \begin{array}{c} 1600 & 17600 \\ 700 & 7700 \end{array} |$ .

Tap for more steps...

Step 3.1

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

$1620 (1600 \cdot 7700 - 700 \cdot 17600) - 675 | \begin{array}{c} 3840 & 17600 \\ 1680 & 7700 \end{array} | + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 3.2

Simplify the determinant.

Tap for more steps...

Step 3.2.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1

Multiply $1600$ by $7700$ .

$1620 (12320000 - 700 \cdot 17600) - 675 | \begin{array}{c} 3840 & 17600 \\ 1680 & 7700 \end{array} | + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 3.2.1.2

Multiply $- 700$ by $17600$ .

$1620 (12320000 - 12320000) - 675 | \begin{array}{c} 3840 & 17600 \\ 1680 & 7700 \end{array} | + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 3.2.2

Subtract $12320000$ from $12320000$ .

$1620 \cdot 0 - 675 | \begin{array}{c} 3840 & 17600 \\ 1680 & 7700 \end{array} | + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 4

Evaluate $| \begin{array}{c} 3840 & 17600 \\ 1680 & 7700 \end{array} |$ .

Tap for more steps...

Step 4.1

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

$1620 \cdot 0 - 675 (3840 \cdot 7700 - 1680 \cdot 17600) + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 4.2

Simplify the determinant.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply $3840$ by $7700$ .

$1620 \cdot 0 - 675 (29568000 - 1680 \cdot 17600) + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 4.2.1.2

Multiply $- 1680$ by $17600$ .

$1620 \cdot 0 - 675 (29568000 - 29568000) + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 4.2.2

Subtract $29568000$ from $29568000$ .

$1620 \cdot 0 - 675 \cdot 0 + 7425 | \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$

Step 5

Evaluate $| \begin{array}{c} 3840 & 1600 \\ 1680 & 700 \end{array} |$ .

Tap for more steps...

Step 5.1

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

$1620 \cdot 0 - 675 \cdot 0 + 7425 (3840 \cdot 700 - 1680 \cdot 1600)$

Step 5.2

Simplify the determinant.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Multiply $3840$ by $700$ .

$1620 \cdot 0 - 675 \cdot 0 + 7425 (2688000 - 1680 \cdot 1600)$

Step 5.2.1.2

Multiply $- 1680$ by $1600$ .

$1620 \cdot 0 - 675 \cdot 0 + 7425 (2688000 - 2688000)$

Step 5.2.2

Subtract $2688000$ from $2688000$ .

$1620 \cdot 0 - 675 \cdot 0 + 7425 \cdot 0$

Step 6

Simplify the determinant.

Tap for more steps...

Step 6.1

Simplify each term.

Tap for more steps...

Step 6.1.1

Multiply $1620$ by $0$ .

$0 - 675 \cdot 0 + 7425 \cdot 0$

Step 6.1.2

Multiply $- 675$ by $0$ .

$0 + 0 + 7425 \cdot 0$

Step 6.1.3

Multiply $7425$ by $0$ .

$0 + 0 + 0$

Step 6.2

Add $0$ and $0$ .

$0 + 0$

Step 6.3

Add $0$ and $0$ .

$0$