The linear system of equations begin{cases} 8 | vedclass

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(D) The given system of equations is a homogeneous system $AX = 0$, where $A = \begin{bmatrix} 8 & -3 & -5 \ 5 & -8 & 3 \ 3 & 5 & -8 \end{bmatrix}$.

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infinitely many non-zero solutions

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(D) The given system of equations is a homogeneous system $AX = 0$, where $A = \begin{bmatrix} 8 & -3 & -5 \ 5 & -8 & 3 \ 3 & 5 & -8 \end{bmatrix}$.To determine the nature of the solutions, we calculate the determinant of the coefficient matrix $D = |A|$.$D = \begin{vmatrix} 8 & -3 & -5 \ 5 & -8 & 3 \ 3 & 5 & -8 \end{vmatrix}$$D = 8((-8)(-8) - (3)(5)) - (-3)((5)(-8) - (3)(3)) + (-5)((5)(5) - (-8)(3))$$D = 8(64 - 15) + 3(-40 - 9) - 5(25 + 24)$$D = 8(49) + 3(-49) - 5(49)$$D = 49(8 - 3 - 5) = 49(0) = 0$.Since the determinant $D = 0$, the system of homogeneous equations has infinitely many non-zero solutions.

The linear system of equations $\begin{cases} 8x - 3y - 5z = 0 \\ 5x - 8y + 3z = 0 \\ 3x + 5y - 8z = 0 \end{cases}$ has

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