The number of 3 times 3 matrices A,whose entr | vedclass

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(A) The system of linear equations is given by $AX = B$,where $A$ is a $3 	imes 3$ matrix,$X = [x, y, z]^T$,and $B = [1, -1, 0]^T$. For a system of l

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$0$

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(A) The system of linear equations is given by $AX = B$,where $A$ is a $3 	imes 3$ matrix,$X = [x, y, z]^T$,and $B = [1, -1, 0]^T$. For a system of linear equations $AX = B$,the number of solutions can be $0$ (inconsistent),$1$ (unique solution),or infinitely many (if the system is consistent and dependent). It is a fundamental property of linear systems that if a system has more than one solution,it must have infinitely many solutions. Therefore,it is impossible for a system of linear equations to have exactly $3$ distinct solutions. Thus,the number of such matrices $A$ is $0$.

The number of $3 \times 3$ matrices $A$,whose entries are either $1$ or $-1$ and for which the system $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$ has exactly three distinct solutions,is

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