The rank of the matrix begin{bmatrix} 2 & -3 | vedclass

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(B) To find the rank of the matrix $A = \begin{bmatrix} 2 & -3 & 4 & 0 \ 5 & -4 & 2 & 1 \ 1 & -3 & 5 & -4 \end{bmatrix}$,we perform row operations t

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$3$

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(B) To find the rank of the matrix $A = \begin{bmatrix} 2 & -3 & 4 & 0 \ 5 & -4 & 2 & 1 \ 1 & -3 & 5 & -4 \end{bmatrix}$,we perform row operations to reduce it to row-echelon form.Step $1$: Swap $R_1$ and $R_3$ to get a $1$ in the first position:$R_1 \leftrightarrow R_3 \implies \begin{bmatrix} 1 & -3 & 5 & -4 \ 5 & -4 & 2 & 1 \ 2 & -3 & 4 & 0 \end{bmatrix}$Step $2$: Eliminate the first column entries below the pivot:$R_2 	o R_2 - 5R_1 \implies \begin{bmatrix} 1 & -3 & 5 & -4 \ 0 & 11 & -23 & 21 \ 2 & -3 & 4 & 0 \end{bmatrix}$$R_3 	o R_3 - 2R_1 \implies \begin{bmatrix} 1 & -3 & 5 & -4 \ 0 & 11 & -23 & 21 \ 0 & 3 & -6 & 8 \end{bmatrix}$Step $3$: Simplify the second and third rows:$R_2 	o R_2 - 3R_3 \implies \begin{bmatrix} 1 & -3 & 5 & -4 \ 0 & 2 & -5 & -3 \ 0 & 3 & -6 & 8 \end{bmatrix}$$R_3 	o 2R_3 - 3R_2 \implies \begin{bmatrix} 1 & -3 & 5 & -4 \ 0 & 2 & -5 & -3 \ 0 & 0 & 3 & 25 \end{bmatrix}$Since there are $3$ non-zero rows in the row-echelon form,the rank of the matrix is $3$.

The rank of the matrix $\begin{bmatrix} 2 & -3 & 4 & 0 \\ 5 & -4 & 2 & 1 \\ 1 & -3 & 5 & -4 \end{bmatrix}$ is

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