The number of non-trivial solutions of the sy | vedclass

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(A) The given system of linear equations is homogeneous,which can be written in matrix form $AX = O$ as: $\begin{bmatrix} 1 & -1 & 1 \ 1 & 2 & -1 \

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$0$

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(A) The given system of linear equations is homogeneous,which can be written in matrix form $AX = O$ as: $\begin{bmatrix} 1 & -1 & 1 \ 1 & 2 & -1 \ 2 & 1 & 3 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}$ To determine the nature of the solutions,we calculate the determinant of the coefficient matrix $A$: $|A| = \begin{vmatrix} 1 & -1 & 1 \ 1 & 2 & -1 \ 2 & 1 & 3 \end{vmatrix}$ $|A| = 1(2 	imes 3 - (-1) 	imes 1) - (-1)(1 	imes 3 - (-1) 	imes 2) + 1(1 	imes 1 - 2 	imes 2)$ $|A| = 1(6 + 1) + 1(3 + 2) + 1(1 - 4)$ $|A| = 7 + 5 - 3 = 9$ Since $|A| 
eq 0$,the matrix $A$ is non-singular and invertible. For a homogeneous system $AX = O$,if $|A| 
eq 0$,the system has only the trivial solution $(x=0, y=0, z=0)$. Therefore,the number of non-trivial solutions is $0$.

The number of non-trivial solutions of the system $x-y+z=0, x+2y-z=0, 2x+y+3z=0$ is

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