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(D) The given system of homogeneous linear equations is:$x + y - z = 0$$3x - y - z = 0$$x - 3y + z = 0$To find the number of solutions,we calculate th

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Infinite

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(D) The given system of homogeneous linear equations is:$x + y - z = 0$$3x - y - z = 0$$x - 3y + z = 0$To find the number of solutions,we calculate the determinant $\Delta$ of the coefficient matrix:$\Delta = \begin{vmatrix} 1 & 1 & -1 \ 3 & -1 & -1 \ 1 & -3 & 1 \end{vmatrix}$Expanding along the first row:$\Delta = 1((-1)(1) - (-1)(-3)) - 1((3)(1) - (-1)(1)) + (-1)((3)(-3) - (-1)(1))$$\Delta = 1(-1 - 3) - 1(3 + 1) - 1(-9 + 1)$$\Delta = 1(-4) - 1(4) - 1(-8)$$\Delta = -4 - 4 + 8 = 0$Since the determinant $\Delta = 0$ for a homogeneous system of equations,the system has non-trivial solutions,which implies there are infinitely many solutions.

The number of solutions of the equations $x + y - z = 0$,$3x - y - z = 0$,and $x - 3y + z = 0$ is

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