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: \(p=4\)

Answer 1

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.