3. Calculate the reduced low echelon form of the following augmented matrix. (5 marks) (1) [} -1&5&3&7 1&-4&-2&8 0&0&1&3 ]

## Step 1: Create the augmented matrix<br />### The given system of equations can be represented by the augmented matrix:<br />$ \left[ \begin{array}{ccc|c} -1 & 5 & 3 & 7 \\ 1 & -4 & -2 & 8 \\ 0 & 0 & 1 & 3 \end{array} \right] $<br /><br />## Step 2: Row operations to achieve row echelon form<br />### Perform row operations to get the matrix into row echelon form.<br />1. $R_1 \rightarrow -R_1$: $ \left[ \begin{array}{ccc|c} 1 & -5 & -3 & -7 \\ 1 & -4 & -2 & 8 \\ 0 & 0 & 1 & 3 \end{array} \right] $<br />2. $R_2 \rightarrow R_2 - R_1$: $ \left[ \begin{array}{ccc|c} 1 & -5 & -3 & -7 \\ 0 & 1 & 1 & 15 \\ 0 & 0 & 1 & 3 \end{array} \right] $<br />3. $R_1 \rightarrow R_1 + 5R_2$: $ \left[ \begin{array}{ccc|c} 1 & 0 & 2 & 68 \\ 0 & 1 & 1 & 15 \\ 0 & 0 & 1 & 3 \end{array} \right] $<br />4. $R_1 \rightarrow R_1 - 2R_3$: $ \left[ \begin{array}{ccc|c} 1 & 0 & 0 & 62 \\ 0 & 1 & 1 & 15 \\ 0 & 0 & 1 & 3 \end{array} \right] $<br />5. $R_2 \rightarrow R_2 - R_3$: $ \left[ \begin{array}{ccc|c} 1 & 0 & 0 & 62 \\ 0 & 1 & 0 & 12 \\ 0 & 0 & 1 & 3 \end{array} \right] $<br /><br />## Step 3: Reduced Row Echelon Form<br />### The matrix is now in reduced row echelon form.