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(C) Consider the matrix $A = \begin{bmatrix} 1 & 4 & -1 \ 2 & 3 & 0 \ 0 & 1 & 2 \end{bmatrix}$.$A$ is a square matrix of order $3 	imes 3$.To find

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$3$

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(C) Consider the matrix $A = \begin{bmatrix} 1 & 4 & -1 \ 2 & 3 & 0 \ 0 & 1 & 2 \end{bmatrix}$.$A$ is a square matrix of order $3 	imes 3$.To find the rank,we calculate the determinant of $A$:$|A| = \begin{vmatrix} 1 & 4 & -1 \ 2 & 3 & 0 \ 0 & 1 & 2 \end{vmatrix}$Expanding along the first row:$|A| = 1(3 	imes 2 - 0 	imes 1) - 4(2 	imes 2 - 0 	imes 0) + (-1)(2 	imes 1 - 3 	imes 0)$$|A| = 1(6 - 0) - 4(4 - 0) - 1(2 - 0)$$|A| = 6 - 16 - 2 = -12$.Since $|A| 
eq 0$,the matrix is non-singular.For a square matrix of order $n$,if the determinant is non-zero,the rank of the matrix is $n$.Therefore,the rank of the given matrix is $3$.

Find the rank of the matrix $\begin{bmatrix} 1 & 4 & -1 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{bmatrix}$.

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