Question #200607

(4.1)Use the reduced row echelon form to determine

2 4 6

0 0 2

2 -1 5


(4.2)Use the reduced row echelon form to determine

1 2 -4

2 -3 1

0 0 2


1
Expert's answer
2021-05-31T15:17:18-0400

4.1--


(246002215)\begin{pmatrix} 2 & 4 & 6 \\ 0 & 0 & 2 \\ 2 & -1 & 5 \end{pmatrix}


(i) R1R12R_1 \rightarrow \dfrac{R_1}{2}


\Rightarrow (123002215)\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 2 \\ 2 & -1 & 5 \end{pmatrix}


(ii) R3R32R1R_3\rightarrow R_3-2R_1

\Rightarrow (123002051)\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 2 \\ 0 & -5 & -1 \end{pmatrix}


(iii) Interchange 2 and 3 row

(123051002)\begin{pmatrix} 1 & 2 & 3 \\ 0 & -5 & -1 \\ 0 & 0 & 2 \end{pmatrix}


(iv) R2R25R_2 \rightarrow \dfrac{R_2}{-5}


(123010.2002)\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 0.2 \\ 0 & 0 & 2 \end{pmatrix}


(v) R3R32R_3 \rightarrow \dfrac{R_3}{2}


(123010.2001)\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 0.2 \\ 0 & 0 & 1 \end{pmatrix}


This is the Row Echelon Form of the matrix.

Since there are 3 non -zero rows hence the Rank is = 3


4.2--


(124231002)\begin{pmatrix} 1 & 2 & -4 \\ 2 & -3 & 1 \\ 0 & 0 & 2 \end{pmatrix}


(i) R2R22R1R_2\rightarrow R_2 -2R_1

\Rightarrow (124079002)\begin{pmatrix} 1 & 2 & -4 \\ 0 & -7 & 9 \\ 0 & 0 & 2 \end{pmatrix}


(ii) R2R27R_2 \rightarrow \dfrac{R_2}{-7} (124019/7002)\begin{pmatrix} 1 & 2 & -4 \\ 0 & -1 & -9/7 \\ 0 & 0 & 2 \end{pmatrix}

\Rightarrow (124019/7002)\begin{pmatrix} 1 & 2 & -4 \\ 0 & -1 & -9/7 \\ 0 & 0 & 2 \end{pmatrix}


(iii) R3R32R_3 \rightarrow \dfrac{R_3}{2}

\Rightarrow (124019/7001)\begin{pmatrix} 1 & 2 & -4 \\ 0 & -1 & -9/7 \\ 0 & 0 & 1 \end{pmatrix}


This is the Row Echelon Form of the matrix.

There are 3 non -zero rows hence the rank of the matrix = 3



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