Finite Math Examples

$[\begin{array}{c} x \\ y \end{array}] \cdot [\begin{array}{c} x - y & x + y \end{array}]$

Step 1

Multiply $[\begin{array}{c} x \\ y \end{array}] [\begin{array}{c} x - y & x + y \end{array}]$ .

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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 $2 \times 1$ and the second matrix is $1 \times 2$ .

Step 1.2

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

$[\begin{array}{c} x (x - y) & x (x + y) \\ y (x - y) & y (x + y) \end{array}]$

Step 1.3

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

$[\begin{array}{c} x^{2} - x y & x^{2} + x y \\ y x - y^{2} & y x + y^{2} \end{array}]$

Step 2

Find the determinant.

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Step 2.1

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

$(x^{2} - x y) (y x + y^{2}) - (y x - y^{2}) (x^{2} + x y)$

Step 2.2

Simplify the determinant.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Expand $(x^{2} - x y) (y x + y^{2})$ using the FOIL Method.

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Step 2.2.1.1.1

Apply the distributive property.

$x^{2} (y x + y^{2}) - x y (y x + y^{2}) - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.1.2

Apply the distributive property.

$x^{2} (y x) + x^{2} y^{2} - x y (y x + y^{2}) - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.1.3

Apply the distributive property.

$x^{2} (y x) + x^{2} y^{2} - x y (y x) - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2

Simplify and combine like terms.

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Step 2.2.1.2.1

Simplify each term.

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Step 2.2.1.2.1.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.2.1.2.1.1.1

Move $x$ .

$x \cdot x^{2} y + x^{2} y^{2} - x y (y x) - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.1.2

Multiply $x$ by $x^{2}$ .

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Step 2.2.1.2.1.1.2.1

Raise $x$ to the power of $1$ .

$x^{1} x^{2} y + x^{2} y^{2} - x y (y x) - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.1.2.2

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

$x^{1 + 2} y + x^{2} y^{2} - x y (y x) - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.1.3

Add $1$ and $2$ .

$x^{3} y + x^{2} y^{2} - x y (y x) - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.2.1.2.1

Move $x$ .

$x^{3} y + x^{2} y^{2} - (x \cdot x) y \cdot y - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.2.2

Multiply $x$ by $x$ .

$x^{3} y + x^{2} y^{2} - x^{2} y \cdot y - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.3

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.2.1.3.1

Move $y$ .

$x^{3} y + x^{2} y^{2} - x^{2} (y \cdot y) - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.3.2

Multiply $y$ by $y$ .

$x^{3} y + x^{2} y^{2} - x^{2} y^{2} - x y \cdot y^{2} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.4

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 2.2.1.2.1.4.1

Move $y^{2}$ .

$x^{3} y + x^{2} y^{2} - x^{2} y^{2} - x (y^{2} y) - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.4.2

Multiply $y^{2}$ by $y$ .

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Step 2.2.1.2.1.4.2.1

Raise $y$ to the power of $1$ .

$x^{3} y + x^{2} y^{2} - x^{2} y^{2} - x (y^{2} y^{1}) - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.4.2.2

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

$x^{3} y + x^{2} y^{2} - x^{2} y^{2} - x y^{2 + 1} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.1.4.3

Add $2$ and $1$ .

$x^{3} y + x^{2} y^{2} - x^{2} y^{2} - x y^{3} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.2

Subtract $x^{2} y^{2}$ from $x^{2} y^{2}$ .

$x^{3} y + 0 - x y^{3} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.2.3

Add $x^{3} y$ and $0$ .

$x^{3} y - x y^{3} - (y x - y^{2}) (x^{2} + x y)$

Step 2.2.1.3

Apply the distributive property.

$x^{3} y - x y^{3} + (- (y x) - - y^{2}) (x^{2} + x y)$

Step 2.2.1.4

Multiply $- - y^{2}$ .

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Step 2.2.1.4.1

Multiply $- 1$ by $- 1$ .

$x^{3} y - x y^{3} + (- y x + 1 y^{2}) (x^{2} + x y)$

Step 2.2.1.4.2

Multiply $y^{2}$ by $1$ .

$x^{3} y - x y^{3} + (- y x + y^{2}) (x^{2} + x y)$

Step 2.2.1.5

Expand $(- y x + y^{2}) (x^{2} + x y)$ using the FOIL Method.

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Step 2.2.1.5.1

Apply the distributive property.

$x^{3} y - x y^{3} - y x (x^{2} + x y) + y^{2} (x^{2} + x y)$

Step 2.2.1.5.2

Apply the distributive property.

$x^{3} y - x y^{3} - y x \cdot x^{2} - y x (x y) + y^{2} (x^{2} + x y)$

Step 2.2.1.5.3

Apply the distributive property.

$x^{3} y - x y^{3} - y x \cdot x^{2} - y x (x y) + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6

Simplify and combine like terms.

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Step 2.2.1.6.1

Simplify each term.

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Step 2.2.1.6.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.2.1.6.1.1.1

Move $x^{2}$ .

$x^{3} y - x y^{3} - y (x^{2} x) - y x (x y) + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6.1.1.2

Multiply $x^{2}$ by $x$ .

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Step 2.2.1.6.1.1.2.1

Raise $x$ to the power of $1$ .

$x^{3} y - x y^{3} - y (x^{2} x^{1}) - y x (x y) + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6.1.1.2.2

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

$x^{3} y - x y^{3} - y x^{2 + 1} - y x (x y) + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6.1.1.3

Add $2$ and $1$ .

$x^{3} y - x y^{3} - y x^{3} - y x (x y) + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.1.6.1.2.1

Move $y$ .

$x^{3} y - x y^{3} - y x^{3} - (y \cdot y) x \cdot x + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6.1.2.2

Multiply $y$ by $y$ .

$x^{3} y - x y^{3} - y x^{3} - y^{2} x \cdot x + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.6.1.3.1

Move $x$ .

$x^{3} y - x y^{3} - y x^{3} - y^{2} (x \cdot x) + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6.1.3.2

Multiply $x$ by $x$ .

$x^{3} y - x y^{3} - y x^{3} - y^{2} x^{2} + y^{2} x^{2} + y^{2} (x y)$

Step 2.2.1.6.1.4

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.2.1.6.1.4.1

Move $y$ .

$x^{3} y - x y^{3} - y x^{3} - y^{2} x^{2} + y^{2} x^{2} + y \cdot y^{2} x$

Step 2.2.1.6.1.4.2

Multiply $y$ by $y^{2}$ .

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Step 2.2.1.6.1.4.2.1

Raise $y$ to the power of $1$ .

$x^{3} y - x y^{3} - y x^{3} - y^{2} x^{2} + y^{2} x^{2} + y^{1} y^{2} x$

Step 2.2.1.6.1.4.2.2

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

$x^{3} y - x y^{3} - y x^{3} - y^{2} x^{2} + y^{2} x^{2} + y^{1 + 2} x$

Step 2.2.1.6.1.4.3

Add $1$ and $2$ .

$x^{3} y - x y^{3} - y x^{3} - y^{2} x^{2} + y^{2} x^{2} + y^{3} x$

Step 2.2.1.6.2

Add $- y^{2} x^{2}$ and $y^{2} x^{2}$ .

$x^{3} y - x y^{3} - y x^{3} + 0 + y^{3} x$

Step 2.2.1.6.3

Add $- y x^{3}$ and $0$ .

$x^{3} y - x y^{3} - y x^{3} + y^{3} x$

Step 2.2.2

Combine the opposite terms in $x^{3} y - x y^{3} - y x^{3} + y^{3} x$ .

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Step 2.2.2.1

Reorder the factors in the terms $x^{3} y$ and $- y x^{3}$ .

$x^{3} y - x y^{3} - x^{3} y + y^{3} x$

Step 2.2.2.2

Subtract $x^{3} y$ from $x^{3} y$ .

$- x y^{3} + 0 + y^{3} x$

Step 2.2.2.3

Add $- x y^{3}$ and $0$ .

$- x y^{3} + y^{3} x$

Step 2.2.2.4

Reorder the factors in the terms $- x y^{3}$ and $y^{3} x$ .

$- y^{3} x + y^{3} x$

Step 2.2.2.5

Add $- y^{3} x$ and $y^{3} x$ .

$0$

Step 3

There is no inverse because the determinant is $0$ .