Reduce the following matrices to row echelon form and row reduced echelon forms: (i)[1p−1217−332](ii)[p−17221−531320] *Find also the ra of these matrices. *Notice: p=4

Step 1 i) Let the matrix be, A=[14−1217−332] Apply R2→R2−2⋅R1 ⇒[14−10−79−332] Apply R3→R3+3⋅R1 ⇒[14−10−79015−1] Step 2 Apply R3→R3+157R2 ⇒[14−10−79001287] Apply R2→−17⋅R2 ⇒[14−101−79001287] Step 3 Apply R3→7128⋅R3 ⇒[14−101−79001] This is the Row Echelon Form of the matrix. Also, the rank of this matrix A is 3. Apply R1→R1−4R2 ⇒[1019901−79001] Step 4 Apply R1→R1−199R3 ⇒[10001−79001] Apply R2→R2+79R3 ⇒[100010001] This is the Reduced Row Echelon Form of the given matrix. Step 5 ii) Let the matrix be represented as, B=[4−17221−531320] Apply R1↔R3 ⇒[132021−534−172] Apply R2→R2−2R1 ⇒[13200−5−934−172] Step 6 Apply R3→R3−4R1 ⇒[13200−5−930−13−12] Apply R2→−15R2

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Kaycee Roche

Answered question

2021-02-22

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

Answer & Explanation

mhalmantus

Skilled2021-02-23Added 105 answers

Step 1
i) Let the matrix be,

A = [\begin{matrix} 1 & 4 & - 1 \\ 2 & 1 & 7 \\ - 3 & 3 & 2 \end{matrix}]

Apply

R_{2} \to R_{2} - 2 \cdot R_{1}

\Rightarrow [\begin{matrix} 1 & 4 & - 1 \\ 0 & - 7 & 9 \\ - 3 & 3 & 2 \end{matrix}]

Apply

R_{3} \to R_{3} + 3 \cdot R_{1}

\Rightarrow [\begin{matrix} 1 & 4 & - 1 \\ 0 & - 7 & 9 \\ 0 & 15 & - 1 \end{matrix}]

Step 2
Apply

R_{3} \to R_{3} + \frac{15}{7} R_{2}

\Rightarrow [\begin{matrix} 1 & 4 & - 1 \\ 0 & - 7 & 9 \\ 0 & 0 & \frac{128}{7} \end{matrix}]

Apply

R_{2} \to - \frac{1}{7} \cdot R_{2}

\Rightarrow [\begin{matrix} 1 & 4 & - 1 \\ 0 & 1 & - \frac{7}{9} \\ 0 & 0 & \frac{128}{7} \end{matrix}]

Step 3
Apply

R_{3} \to \frac{7}{128} \cdot R_{3}

\Rightarrow [\begin{matrix} 1 & 4 & - 1 \\ 0 & 1 & - \frac{7}{9} \\ 0 & 0 & 1 \end{matrix}]

This is the Row Echelon Form of the matrix.
Also, the rank of this matrix A is 3.
Apply

R_{1} \to R_{1} - 4 R_{2}

\Rightarrow [\begin{matrix} 1 & 0 & \frac{19}{9} \\ 0 & 1 & - \frac{7}{9} \\ 0 & 0 & 1 \end{matrix}]

Step 4
Apply

R_{1} \to R_{1} - \frac{19}{9} R_{3}

\Rightarrow [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & - \frac{7}{9} \\ 0 & 0 & 1 \end{matrix}]

Apply

R_{2} \to R_{2} + \frac{7}{9} R_{3}

\Rightarrow [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

This is the Reduced Row Echelon Form of the given matrix.
Step 5
ii) Let the matrix be represented as,

B = [\begin{matrix} 4 & - 1 & 7 & 2 \\ 2 & 1 & - 5 & 3 \\ 1 & 3 & 2 & 0 \end{matrix}]

Apply

R_{1} \leftrightarrow R_{3}

\Rightarrow [\begin{matrix} 1 & 3 & 2 & 0 \\ 2 & 1 & - 5 & 3 \\ 4 & - 1 & 7 & 2 \end{matrix}]

Apply

R_{2} \to R_{2} - 2 R_{1}

\Rightarrow [\begin{matrix} 1 & 3 & 2 & 0 \\ 0 & - 5 & - 9 & 3 \\ 4 & - 1 & 7 & 2 \end{matrix}]

Step 6
Apply

R_{3} \to R_{3} - 4 R_{1}

\Rightarrow [\begin{matrix} 1 & 3 & 2 & 0 \\ 0 & - 5 & - 9 & 3 \\ 0 & - 13 & - 1 & 2 \end{matrix}]

Apply

R_{2} \to \frac{- 1}{5} R_{2}

This is helpful 4 Flag

Jeffrey Jordon

Expert2022-01-23Added 2605 answers

Answer is given below (on video)

Jeffrey Jordon

Expert2022-08-23Added 2605 answers

Answer is given below (on video)

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