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Reduce the following matrices to their Echelon forms and reduced echelon form
$$\left(\begin{array}{ccc}{-1} & {3} & {-5} \\ {1} & {-1} & {3}\end{array}\right)$$

$$\left(\begin{array}{ccc}{-1} & {3} & {-5} \\ {1} & {-1} & {3}\end{array}\right) R_{1} \rightarrow-R_{1} $$

$$\left(\begin{array}{ccc}{1} & {-3} & {5} \\ {1} & {-1} & {3}\end{array}\right) {R_{2} \rightarrow R_{2}-R_{1}} $$

$$\left(\begin{array}{ccc}{1} & {-3} & {5} \\ {0} & {2} & {-2}\end{array}\right) R_{2} \rightarrow 1 / 2 R_{2} $$

$$\left(\begin{array}{rrr}{1} & {-3} & {5} \\ {0} & {1} & {-1}\end{array}\right)$$

Row echelon form

$$\left(\begin{array}{ccc}{1} & {-3} & {5} \\ {0} & {1} & {-1}\end{array}\right) {R}_{1} \rightarrow R_{1}+3 R_{2} $$

$$\left(\begin{array}{ccc}{1} & {0} & {2} \\ {0} & {1} & {-1}\end{array}\right)$$

Reduced echelon form

Reduce the following matrices to their Echelon forms and reduced echelon form
$$\left(\begin{array}{cccc}{2} & {-1} & {4} & {-1} \\ {1} & {3} & {0} & {5} \\ {1} & {1} & {1} & {2}\end{array}\right)$$

$$\left(\begin{array}{cccc} {2} & {-1} & {4} & {-1} \\ {1} & {3} & {0} & {5} \\ {1} & {1} & {1} & {2}\end{array}\right) {R_{1} \leftrightarrow R_{3}} $$

$$\left(\begin{array}{cccc}{1} & {1} & {1} & {2} \\ {1} & {3} & {0} & {5} \\ {2} & {-1} & {4} & {-1}\end{array}\right)\begin{array}{ll}{ R_{2} \rightarrow R_{2}-R_{1}} \\{R_{3} \rightarrow R_{3}-2 R_{1}}\end{array} $$

$$\left(\begin{array}{cccc}{1} & {1} & {1} & {2} \\ {0} & {2} & {-1} & {3} \\ {0} & {-3} & {2} & {-5}\end{array}\right) R_{2} \rightarrow 1/2 R_{2} $$

$$\left(\begin{array}{cccc}{1} & {1} & {1} & {2} \\ {0} & {1} & {-1 / 2} & {3 / 2} \\ {0} & {-3} & {2} & {-5}\end{array}\right) R_{3} \rightarrow R_{3}+3 R_{2} $$

$$\left(\begin{array}{cccc}{1} & {1} & {1} & {2} \\ {0} & {1} & {-1 / 2} & {3 / 2} \\ {0} & {0} & {1 / 2} & {-1 / 2}\end{array}\right) R_{3} \rightarrow 2 R_{3} $$

$$\left(\begin{array}{cccc}{1} & {1} & {1} & {2} \\ {0} & {1} & {-1 / 2} & {3 / 2} \\ {0} & {0} & {1} & {-1}\end{array}\right)$$

Row echelon form

$$\left(\begin{array}{cccc}{1} & {1} & {1} & {2} \\ {0} & {1} & {-1 / 2} & {3 / 2} \\ {0} & {0} & {1} & {-1}\end{array}\right) \begin{array}{l}{R_{1} \rightarrow R_{1} -R_{3}} \\ {R_{2} \rightarrow R_{2}+1 / 2 R_{3}}\end{array} $$

$$\left(\begin{array}{cccc}{1} & {1} & {0} & {3} \\ {0} & {1} & {0} & {1} \\ {0} & {0} & {1} & {-1}\end{array}\right) R_{1} \rightarrow R_{1}-R_{2} $$

$$\left(\begin{array}{cccc}{1} & {0} & {0} & {2} \\ {0} & {1} & {0} & {1} \\ {0} & {0} & {1} & {-1}\end{array}\right)$$

reduced echelon form

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