Finite Math Examples

⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

Step 1

Write the matrix as a product of a lower triangular matrix and an upper triangular matrix.

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 l_{21} & 1 & 0 l_{31} & l_{32} & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} u_{11} & u_{12} & u_{13} 0 & u_{22} & u_{23} 0 & 0 & u_{33} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

Step 2

Multiply

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 l_{21} & 1 & 0 l_{31} & l_{32} & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} u_{11} & u_{12} & u_{13} 0 & u_{22} & u_{23} 0 & 0 & u_{33} \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}] [\begin{array}{c} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}]$ .

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Step 2.1

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

$3 \times 3$ and the second matrix is

3 \times 3

$3 \times 3$ .

Step 2.2

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

⎡ ⎢ ⎣ \begin{matrix} 1 u_{11} + 0 \cdot 0 + 0 \cdot 0 & 1 u_{12} + 0 u_{22} + 0 \cdot 0 & 1 u_{13} + 0 u_{23} + 0 u_{33} l_{21} u_{11} + 1 \cdot 0 + 0 \cdot 0 & l_{21} u_{12} + 1 u_{22} + 0 \cdot 0 & l_{21} u_{13} + 1 u_{23} + 0 u_{33} l_{31} u_{11} + l_{32} \cdot 0 + 1 \cdot 0 & l_{31} u_{12} + l_{32} u_{22} + 1 \cdot 0 & l_{31} u_{13} + l_{32} u_{23} + 1 u_{33} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 u_{11} + 0 \cdot 0 + 0 \cdot 0 & 1 u_{12} + 0 u_{22} + 0 \cdot 0 & 1 u_{13} + 0 u_{23} + 0 u_{33} \\ l_{21} u_{11} + 1 \cdot 0 + 0 \cdot 0 & l_{21} u_{12} + 1 u_{22} + 0 \cdot 0 & l_{21} u_{13} + 1 u_{23} + 0 u_{33} \\ l_{31} u_{11} + l_{32} \cdot 0 + 1 \cdot 0 & l_{31} u_{12} + l_{32} u_{22} + 1 \cdot 0 & l_{31} u_{13} + l_{32} u_{23} + 1 u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

Step 2.3

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

⎡ ⎢ ⎣ \begin{matrix} u_{11} & u_{12} & u_{13} l_{21} u_{11} & l_{21} u_{12} + u_{22} & l_{21} u_{13} + u_{23} l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + u_{33} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} u_{11} & u_{12} & u_{13} \\ l_{21} u_{11} & l_{21} u_{12} + u_{22} & l_{21} u_{13} + u_{23} \\ l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + u_{33} \end{array}] = [\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} u_{11} & u_{12} & u_{13} l_{21} u_{11} & l_{21} u_{12} + u_{22} & l_{21} u_{13} + u_{23} l_{31} u_{11} & l_{31} u_{12} + l_{32} u_{22} & l_{31} u_{13} + l_{32} u_{23} + u_{33} \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 3 & 1 & - 3 1 & - 2 & - 5 \end{matrix} ⎤ ⎥ ⎦

Step 3

Solve.

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Step 3.1

Write as a linear system of equations.

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{11} = 3

$l_{21} u_{11} = 3$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Step 3.2

Solve the system of equations.

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Step 3.2.1

Replace all occurrences of

u_{11}

$u_{11}$ with

1

$1$ in each equation.

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Step 3.2.1.1

Replace all occurrences of

u_{11}

$u_{11}$ in

l_{21} u_{11} = 3

$l_{21} u_{11} = 3$ with

1

$1$ .

l_{21} \cdot 1 = 3

$l_{21} \cdot 1 = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Step 3.2.1.2

Simplify the left side.

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Step 3.2.1.2.1

Multiply

l_{21}

$l_{21}$ by

1

$1$ .

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Step 3.2.1.3

Replace all occurrences of

u_{11}

$u_{11}$ in

l_{31} u_{11} = 1

$l_{31} u_{11} = 1$ with

1

$1$ .

l_{31} \cdot 1 = 1

$l_{31} \cdot 1 = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Step 3.2.1.4

Simplify the left side.

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Step 3.2.1.4.1

Multiply

l_{31}

$l_{31}$ by

1

$1$ .

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

l_{31} = 1

$l_{31} = 1$

l_{21} = 3

$l_{21} = 3$

u_{11} = 1

$u_{11} = 1$

u_{12} = 1

$u_{12} = 1$

u_{13} = 1

$u_{13} = 1$

l_{21} u_{12} + u_{22} = 1

$l_{21} u_{12} + u_{22} = 1$

l_{21} u_{13} + u_{23} = - 3

$l_{21} u_{13} + u_{23} = - 3$

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$

l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Step 3.2.2

Replace all occurrences of

l_{31}

$l_{31}$ with

1

$1$ in each equation.

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Step 3.2.2.1

Replace all occurrences of

l_{31}

$l_{31}$ in

l_{31} u_{12} + l_{32} u_{22} = - 2

$l_{31} u_{12} + l_{32} u_{22} = - 2$ with

$1$ .

$1 \cdot u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Multiply

$u_{12}$ by

$1$ .

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$

Step 3.2.2.3

Replace all occurrences of

$l_{31}$ in

$l_{31} u_{13} + l_{32} u_{23} + u_{33} = - 5$ with

$1$ .

$1 \cdot u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

Step 3.2.2.4

Simplify the left side.

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Step 3.2.2.4.1

Multiply

$u_{13}$ by

$1$ .

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{12} + u_{22} = 1$

$l_{21} u_{13} + u_{23} = - 3$

Step 3.2.3

Replace all occurrences of

$l_{21}$ with

$3$ in each equation.

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Step 3.2.3.1

Replace all occurrences of

$l_{21}$ in

$l_{21} u_{12} + u_{22} = 1$ with

$3$ .

$3 \cdot u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{13} + u_{23} = - 3$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Multiply

$3$ by

$u_{12}$ .

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{21} u_{13} + u_{23} = - 3$

Step 3.2.3.3

Replace all occurrences of

$l_{21}$ in

$l_{21} u_{13} + u_{23} = - 3$ with

$3$ .

$3 \cdot u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.3.4

Simplify the left side.

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Step 3.2.3.4.1

Multiply

$3$ by

$u_{13}$ .

$3 u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$3 u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$3 u_{13} + u_{23} = - 3$

$3 u_{12} + u_{22} = 1$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.4

Replace all occurrences of

$u_{12}$ with

$1$ in each equation.

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Step 3.2.4.1

Replace all occurrences of

$u_{12}$ in

$3 u_{12} + u_{22} = 1$ with

$1$ .

$3 (1) + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Multiply

$3$ by

$1$ .

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$u_{12} + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.4.3

Replace all occurrences of

$u_{12}$ in

$u_{12} + l_{32} u_{22} = - 2$ with

$1$ .

$1 + l_{32} u_{22} = - 2$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.4.4

Simplify the left side.

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Step 3.2.4.4.1

Remove parentheses.

$1 + l_{32} u_{22} = - 2$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 + l_{32} u_{22} = - 2$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 + l_{32} u_{22} = - 2$

$3 + u_{22} = 1$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.5

Replace all occurrences of

$u_{13}$ with

$1$ in each equation.

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Step 3.2.5.1

Move all terms not containing

$u_{22}$ to the right side of the equation.

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Step 3.2.5.1.1

Subtract

$3$ from both sides of the equation.

$u_{22} = 1 - 3$

$1 + l_{32} u_{22} = - 2$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.5.1.2

Subtract

$3$ from

$1$ .

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$3 u_{13} + u_{23} = - 3$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.5.2

Replace all occurrences of

$u_{13}$ in

$3 u_{13} + u_{23} = - 3$ with

$1$ .

$3 (1) + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.5.3

Simplify the left side.

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Step 3.2.5.3.1

Multiply

$3$ by

$1$ .

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$u_{13} + l_{32} u_{23} + u_{33} = - 5$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.5.4

Replace all occurrences of

$u_{13}$ in

$u_{13} + l_{32} u_{23} + u_{33} = - 5$ with

$1$ .

$1 + l_{32} u_{23} + u_{33} = - 5$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.5.5

Simplify the left side.

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Step 3.2.5.5.1

Remove parentheses.

$1 + l_{32} u_{23} + u_{33} = - 5$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 + l_{32} u_{23} + u_{33} = - 5$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 + l_{32} u_{23} + u_{33} = - 5$

$3 + u_{23} = - 3$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.6

Replace all occurrences of

$u_{22}$ with

$- 2$ in each equation.

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Step 3.2.6.1

Move all terms not containing

$u_{23}$ to the right side of the equation.

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Step 3.2.6.1.1

Subtract

$3$ from both sides of the equation.

$u_{23} = - 3 - 3$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.6.1.2

Subtract

$3$ from

$- 3$ .

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$1 + l_{32} u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.6.2

Replace all occurrences of

$u_{22}$ in

$1 + l_{32} u_{22} = - 2$ with

$- 2$ .

$1 + l_{32} \cdot - 2 = - 2$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.6.3

Simplify the left side.

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Step 3.2.6.3.1

Move

$- 2$ to the left of

$l_{32}$ .

$1 - 2 l_{32} = - 2$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 2 l_{32} = - 2$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 2 l_{32} = - 2$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.7

Replace all occurrences of

$u_{23}$ with

$- 6$ in each equation.

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Step 3.2.7.1

Move all terms not containing

$l_{32}$ to the right side of the equation.

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Step 3.2.7.1.1

Subtract

$1$ from both sides of the equation.

$- 2 l_{32} = - 2 - 1$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.7.1.2

Subtract

$1$ from

$- 2$ .

$- 2 l_{32} = - 3$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 2 l_{32} = - 3$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.7.2

Divide each term in

$- 2 l_{32} = - 3$ by

$- 2$ and simplify.

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Step 3.2.7.2.1

Divide each term in

$- 2 l_{32} = - 3$ by

$- 2$ .

$\frac{- 2 l_{32}}{- 2} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.7.2.2

Simplify the left side.

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Step 3.2.7.2.2.1

Cancel the common factor of

$- 2$ .

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Step 3.2.7.2.2.1.1

Cancel the common factor.

$\frac{- 2 l_{32}}{- 2} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.7.2.2.1.2

Divide

$l_{32}$ by

$1$ .

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{- 3}{- 2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.7.2.3

Simplify the right side.

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Step 3.2.7.2.3.1

Dividing two negative values results in a positive value.

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$1 + l_{32} u_{23} + u_{33} = - 5$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.7.3

Replace all occurrences of

$u_{23}$ in

$1 + l_{32} u_{23} + u_{33} = - 5$ with

$- 6$ .

$1 + l_{32} \cdot - 6 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.7.4

Simplify the left side.

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Step 3.2.7.4.1

Move

$- 6$ to the left of

$l_{32}$ .

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 6 l_{32} + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.8

Replace all occurrences of

$l_{32}$ with

$\frac{3}{2}$ in each equation.

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Step 3.2.8.1

Replace all occurrences of

$l_{32}$ in

$1 - 6 l_{32} + u_{33} = - 5$ with

$\frac{3}{2}$ .

$1 - 6 (\frac{3}{2}) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.8.2

Simplify the left side.

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Step 3.2.8.2.1

Simplify

$1 - 6 (\frac{3}{2}) + u_{33}$ .

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Step 3.2.8.2.1.1

Simplify each term.

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Step 3.2.8.2.1.1.1

Cancel the common factor of

$2$ .

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Step 3.2.8.2.1.1.1.1

Factor

$2$ out of

$- 6$ .

$1 + 2 (- 3) (\frac{3}{2}) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.8.2.1.1.1.2

Cancel the common factor.

$1 + 2 \cdot (- 3 (\frac{3}{2})) + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.8.2.1.1.1.3

Rewrite the expression.

$1 - 3 \cdot 3 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 3 \cdot 3 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.8.2.1.1.2

Multiply

$- 3$ by

$3$ .

$1 - 9 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$1 - 9 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.8.2.1.2

Subtract

$9$ from

$1$ .

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$- 8 + u_{33} = - 5$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.9

Move all terms not containing

$u_{33}$ to the right side of the equation.

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Step 3.2.9.1

Add

$8$ to both sides of the equation.

$u_{33} = - 5 + 8$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.9.2

Add

$- 5$ and

$8$ .

$u_{33} = 3$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

$u_{33} = 3$

$l_{32} = \frac{3}{2}$

$u_{23} = - 6$

$u_{22} = - 2$

$l_{31} = 1$

$l_{21} = 3$

$u_{11} = 1$

$u_{12} = 1$

$u_{13} = 1$

Step 3.2.10

Solve the system of equations.

$u_{33} = 3 l_{32} = \frac{3}{2} u_{23} = - 6 u_{22} = - 2 l_{31} = 1 l_{21} = 3 u_{11} = 1 u_{12} = 1 u_{13} = 1$

Step 3.2.11

List all of the solutions.

$u_{33} = 3, l_{32} = \frac{3}{2}, u_{23} = - 6, u_{22} = - 2, l_{31} = 1, l_{21} = 3, u_{11} = 1, u_{12} = 1, u_{13} = 1$

Step 4

Substitute in the solved values.

$[\begin{array}{c} 1 & 1 & 1 \\ 3 & 1 & - 3 \\ 1 & - 2 & - 5 \end{array}] = [\begin{array}{c} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 1 & \frac{3}{2} & 1 \end{array}] [\begin{array}{c} 1 & 1 & 1 \\ 0 & - 2 & - 6 \\ 0 & 0 & 3 \end{array}]$

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