Question

# Reduce the following matrices to row echelon form and row reduced echelon forms:(i)begin{bmatrix}1 & p & -1 2 & 1 & 7 -3 & 3 & 2 end{bmatrix}(ii) begin{bmatrix}p & -1 & 7&2 2 & 1 & -5 & 3 1 & 3 & 2 & 0 end{bmatrix}*Find also the ra of these matrices.*Notice:p=4

Matrices

Reduce the following matrices to row echelon form and row reduced echelon forms:
$$(i)\begin{bmatrix}1 & p & -1 \\ 2 & 1 & 7 \\ -3 & 3 & 2 \end{bmatrix} (ii) \begin{bmatrix}p & -1 & 7&2 \\ 2 & 1 & -5 & 3 \\ 1 & 3 & 2 & 0 \end{bmatrix}$$
*Find also the ra
of these matrices.
*Notice: $$p=4$$

2021-02-23
Step 1
i) Let the matrix be, $$A=\begin{bmatrix}1 & 4 & -1 \\ 2 & 1 & 7 \\ -3 & 3 & 2 \end{bmatrix}$$
Apply $$R_2 \rightarrow R_2-2 \cdot R_1$$
$$\Rightarrow \begin{bmatrix}1 & 4 & -1 \\ 0 & -7& 9 \\ -3 & 3 & 2 \end{bmatrix}$$ Apply $$R_3 \rightarrow R_3+3 \cdot R_1$$
$$\Rightarrow \begin{bmatrix}1 & 4 & -1 \\ 0 & -7& 9 \\ 0 & 15 & -1 \end{bmatrix}$$
Step 2
Apply $$R_3 \rightarrow R_3 + \frac{15}{7}R_2$$
$$\Rightarrow \begin{bmatrix}1 & 4 & -1 \\ 0 & -7& 9 \\ 0 & 0 & \frac{128}{7} \end{bmatrix}$$
Apply $$R_2 \rightarrow -\frac{1}{7} \cdot R_2$$
$$\Rightarrow \begin{bmatrix}1 & 4 & -1 \\ 0 & 1& -\frac{7}{9} \\ 0 & 0 & \frac{128}{7} \end{bmatrix}$$
Step 3
Apply $$R_3 \rightarrow \frac{7}{128} \cdot R_3$$
$$\Rightarrow \begin{bmatrix}1 & 4 & -1 \\ 0 & 1& -\frac{7}{9} \\ 0 & 0 & 1 \end{bmatrix}$$
This is the Row Echelon Form of the matrix.
Also, the rank of this matrix A is 3.
Apply $$R_1 \rightarrow R_1-4R_2$$
$$\Rightarrow \begin{bmatrix}1 & 0 & \frac{19}{9} \\ 0 & 1& -\frac{7}{9} \\ 0 & 0 & 1 \end{bmatrix}$$
Step 4
Apply $$R_1 \rightarrow R_1 - \frac{19}{9} R_3$$
$$\Rightarrow \begin{bmatrix}1 & 0 &0 \\ 0 & 1& -\frac{7}{9} \\ 0 & 0 & 1 \end{bmatrix}$$
Apply $$R_2 \rightarrow R_2+\frac{7}{9}R_3$$
$$\Rightarrow \begin{bmatrix}1 & 0 &0 \\ 0 & 1& 0 \\ 0 & 0 & 1 \end{bmatrix}$$
This is the Reduced Row Echelon Form of the given matrix.
Step 5
ii) Let the matrix be represented as, $$B=\begin{bmatrix}4 & -1 & 7&2 \\ 2 & 1 & -5 & 3 \\ 1 & 3 & 2 & 0 \end{bmatrix}$$
Apply $$R_1 \leftrightarrow R_3$$
$$\Rightarrow \begin{bmatrix}1 & 3 & 2&0 \\ 2 & 1 & -5 & 3 \\ 4 & -1 & 7 & 2 \end{bmatrix}$$
Apply $$R_2 \rightarrow R_2-2R_1$$
$$\Rightarrow \begin{bmatrix}1 & 3 & 2&0 \\ 0 & -5 & -9 & 3 \\ 4 & -1 & 7 & 2 \end{bmatrix}$$
Step 6
Apply $$R_3 \rightarrow R_3-4R_1$$
$$\Rightarrow \begin{bmatrix}1 & 3 & 2&0 \\ 0 & -5 & -9 & 3 \\ 0 & -13 & -1 & 2 \end{bmatrix}$$
Apply $$R_2 \rightarrow \frac{-1}{5}R_2$$
$$\Rightarrow \begin{bmatrix}1 & 3 & 2&0 \\ 0 & 1 & \frac{9}{5} & \frac{-3}{5} \\ 0 & -13 & -1 & 2 \end{bmatrix}$$
Apply $$R_3 \rightarrow R_3 + 13R_2$$
$$\Rightarrow \begin{bmatrix}1 & 3 & 2&0 \\ 0 & 1 & \frac{9}{5} & \frac{-3}{5} \\ 0 & 0 & \frac{112}{5} & \frac{-29}{5} \end{bmatrix}$$
Step 7
Apply $$R_1 \rightarrow R_1 - 3R_2$$
$$\Rightarrow \begin{bmatrix}1 & 0 & \frac{-17}{5}&\frac{9}{5} \\ 0 & 1 & \frac{9}{5} & \frac{-3}{5} \\ 0 & 0 & \frac{112}{5} & \frac{-29}{5} \end{bmatrix}$$
Apply $$R_3 \rightarrow \frac{5}{112}R_3$$
$$\Rightarrow \begin{bmatrix}1 & 0 & \frac{-17}{5}&\frac{9}{5} \\ 0 & 1 & \frac{9}{5} & \frac{-3}{5} \\ 0 & 0 & 1 & \frac{-29}{5} \end{bmatrix}$$
This is the Row Echelon Form of the matrix.
Step 8
Also, the rank of the matrix B is 3.
Apply $$R_2 \rightarrow R_2-\frac{9}{5}R_3$$
$$\Rightarrow \begin{bmatrix}1 & 0 & \frac{-17}{5}&\frac{9}{5} \\ 0 & 1 & 0 & \frac{-15}{112} \\ 0 & 0 & 1 & \frac{-29}{5} \end{bmatrix}$$
Apply $$R_1 \rightarrow R_1 + \frac{17}{5}R_3$$
$$\Rightarrow \begin{bmatrix}1 & 0 & 0&\frac{103}{112} \\ 0 & 1 & 0 & \frac{-15}{112} \\ 0 & 0 & 1 & \frac{-29}{5} \end{bmatrix}$$
This is the Reduced Row Echelon Form of the given matrix B.