The rank of the matrix begin{bmatrix} 3 & 5 & | vedclass

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(B) Let $A = \begin{bmatrix} 3 & 5 & -1 & 4 \ 2 & 1 & 3 & -2 \ 8 & 11 & 1 & 6 \ -7 & -14 & 6 & -14 \end{bmatrix}$.Applying row operations to reduce

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) Let $A = \begin{bmatrix} 3 & 5 & -1 & 4 \ 2 & 1 & 3 & -2 \ 8 & 11 & 1 & 6 \ -7 & -14 & 6 & -14 \end{bmatrix}$.Applying row operations to reduce the matrix to row-echelon form:$R_1 ightarrow R_1 - R_2$ gives $\begin{bmatrix} 1 & 4 & -4 & 6 \ 2 & 1 & 3 & -2 \ 8 & 11 & 1 & 6 \ -7 & -14 & 6 & -14 \end{bmatrix}$.Applying $R_2 ightarrow R_2 - 2R_1$, $R_3 ightarrow R_3 - 8R_1$, $R_4 ightarrow R_4 + 7R_1$:$\begin{bmatrix} 1 & 4 & -4 & 6 \ 0 & -7 & 11 & -14 \ 0 & -21 & 33 & -42 \ 0 & 14 & -22 & 28 \end{bmatrix}$.Applying $R_3 ightarrow R_3 - 3R_2$:$\begin{bmatrix} 1 & 4 & -4 & 6 \ 0 & -7 & 11 & -14 \ 0 & 0 & 0 & 0 \ 0 & 14 & -22 & 28 \end{bmatrix}$.Applying $R_3 \leftrightarrow R_4$:$\begin{bmatrix} 1 & 4 & -4 & 6 \ 0 & -7 & 11 & -14 \ 0 & 14 & -22 & 28 \ 0 & 0 & 0 & 0 \end{bmatrix}$.Applying $R_3 ightarrow R_3 + 2R_2$:$\begin{bmatrix} 1 & 4 & -4 & 6 \ 0 & -7 & 11 & -14 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \end{bmatrix}$.The number of non-zero rows is $2$.Therefore, the rank of the matrix is $2$.

The rank of the matrix $\begin{bmatrix} 3 & 5 & -1 & 4 \\ 2 & 1 & 3 & -2 \\ 8 & 11 & 1 & 6 \\ -7 & -14 & 6 & -14 \end{bmatrix}$ is

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