Find the rank of the matrix by reducing into normal form

$$tm = \begin{matrix} 7 & 1 &2 & 3 & 5 \\ 0 & 1 & 0 & 2 & 3 \\ -4 & 6 & -1 & 2 & 8 \\ 1& 5 & 0 & -3 &0 \\ \end{matrix}$$

The given matrix

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & -4 & 6 & -1 & 2 & 8 \\ \hline 4 & 1 & 5 & 0 & -3 & 0 \\ \hline \end{array}$

Divide the 1 st row by 7

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 1 & 1 / 7 & 2 / 7 & 3 / 7 & 5 / 7 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & -4 & 6 & -1 & 2 & 8 \\ \hline 4 & 1 & 5 & 0 & -3 & 0 \\ \hline \end{array}$

Multiply the 1st row by -4

$\begin{array}{|c|c|c|c|c|c|} \hline & A_{1} & A_{2} & A_{3} & A_{4} & A_{5} \\ \hline 1 & -4 & -4 / 7 & -8 / 7 & -12 / 7 & -20 / 7 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & -4 & 6 & -1 & 2 & 8 \\ \hline 4 & 1 & 5 & 0 & -3 & 0 \\ \hline \end{array}$

Subtract the 1st row from the 3 rd row and restore it

$\begin{array}{|c|c|c|c|c|c|} \hline & A_{1} & A_{2} & A_{3} & A_{4} & A_{5} \\ \hline 1 & 1 & 1 / 7 & 2 / 7 & 3 / 7 & 5 / 7 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 46 / 7 & 1 / 7 & 26 / 7 & 76 / 7 \\ \hline 4 & 1 & 5 & 0 & -3 & 0 \\ \hline \end{array}$

Subtract the 1st row from the 4th row

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 1 & 1 / 7 & 2 / 7 & 3 / 7 & 5 / 7 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 46 / 7 & 1 / 7 & 26 / 7 & 76 / 7 \\ \hline 4 & 0 & 34 / 7 & -2 / 7 & -24 / 7 & -5 / 7 \\ \hline \end{array}$

Restore the 1st row to the original view

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 46 / 7 & 1 / 7 & 26 / 7 & 76 / 7 \\ \hline 4 & 0 & 34 / 7 & -2 / 7 & -24 / 7 & -5 / 7 \\ \hline \end{array}$

Multiply the 2 nd row by 46 / 7

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 46 / 7 & 0 & 92 / 7 & 138 / 7 \\ \hline 3 & 0 & 46 / 7 & 1 / 7 & 26 / 7 & 76 / 7 \\ \hline 4 & 0 & 34 / 7 & -2 / 7 & -24 / 7 & -5 / 7 \\ \hline \end{array}$

Subtract the 2nd row from the 3rd row and restore it

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 0 & 1 / 7 & -66 / 7 & -62 / 7 \\ \hline 4 & 0 & 34 / 7 & -2 / 7 & -24 / 7 & -5 / 7 \\ \hline \end{array}$

Multiply the 2nd row by 34 / 7

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 34 / 7 & 0 & 68 / 7 & 102 / 7 \\ \hline 3 & 0 & 0 & 1 / 7 & -66 / 7 & -62 / 7 \\ \hline 4 & 0 & 34 / 7 & -2 / 7 & -24 / 7 & -5 / 7 \\ \hline \end{array}$

Subtract the 2 nd row from the 4 th row and restore it

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 0 & 1 / 7 & -66 / 7 & -62 / 7 \\ \hline 4 & 0 & 0 & -2 / 7 & -92 / 7 & -107 / 7 \\ \hline \end{array}$

Divide the 3 rd row by 1 / 7

$\begin{array}{|c|c|c|c|c|c|} \hline & A_{1} & A_{2} & A_{3} & A_{4} & A_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 0 & 1 & -66 & -62 \\ \hline 4 & 0 & 0 & -2 / 7 & -92 / 7 & -107 / 7 \\ \hline \end{array}$

Multiply the 3 rd row by -2 / 7

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 0 & -2 / 7 & 132 / 7 & 124 / 7 \\ \hline 4 & 0 & 0 & -2 / 7 & -92 / 7 & -107 / 7 \\ \hline \end{array}$

Subtract the 3rd row from the 4 th row and restore it

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 0 & 1 & -66 & -62 \\ \hline 4 & 0 & 0 & 0 & -32 & -33 \\ \hline \end{array}$

Restore the 3rd row to the original view

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 0 & 1 / 7 & -66 / 7 & -62 / 7 \\ \hline 4 & 0 & 0 & 0 & -32 & -33 \\ \hline \end{array}$

Calculate the number of linearly independent rows

$\begin{array}{|c|c|c|c|c|c|} \hline & \mathrm{A}_{1} & \mathrm{~A}_{2} & \mathrm{~A}_{3} & \mathrm{~A}_{4} & \mathrm{~A}_{5} \\ \hline 1 & 7 & 1 & 2 & 3 & 5 \\ \hline 2 & 0 & 1 & 0 & 2 & 3 \\ \hline 3 & 0 & 0 & 1 / 7 & -66 / 7 & -62 / 7 \\ \hline 4 & 0 & 0 & 0 & -32 & -33 \\ \hline \end{array}$

$\therefore$ Matrix rank is 4