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(C) Given matrix $A = \begin{bmatrix} 1 & 1 & -1 & 0 \ 4 & 4 & -3 & 1 \ b & 2 & 2 & 2 \ 9 & 9 & b & 3 \end{bmatrix}$.For the rank of the matrix to

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-$6$

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(C) Given matrix $A = \begin{bmatrix} 1 & 1 & -1 & 0 \ 4 & 4 & -3 & 1 \ b & 2 & 2 & 2 \ 9 & 9 & b & 3 \end{bmatrix}$.For the rank of the matrix to be $3$,the determinant of the matrix must be $0$,and there must exist at least one non-zero minor of order $3$.Applying row operations to simplify the matrix:$R_2 ightarrow R_2 - 4R_1$ and $R_4 ightarrow R_4 - 9R_1$:$A \sim \begin{bmatrix} 1 & 1 & -1 & 0 \ 0 & 0 & 1 & 1 \ b & 2 & 2 & 2 \ 0 & 0 & b+9 & 3 \end{bmatrix}$.Applying $R_4 ightarrow R_4 - 3R_2$:$A \sim \begin{bmatrix} 1 & 1 & -1 & 0 \ 0 & 0 & 1 & 1 \ b & 2 & 2 & 2 \ 0 & 0 & b+6 & 0 \end{bmatrix}$.For the rank to be $3$,the fourth row must be a linear combination of the other rows,which implies the last row must become zero after further reduction,or the determinant must be zero.Looking at the structure,if $b+6 = 0$,then $b = -6$.Substituting $b = -6$ into the matrix,the row operations confirm the rank is $3$.

$A$ value of $b$ for which the rank of the matrix $A = \begin{bmatrix} 1 & 1 & -1 & 0 \\ 4 & 4 & -3 & 1 \\ b & 2 & 2 & 2 \\ 9 & 9 & b & 3 \end{bmatrix}$ is $3$,is

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