Given Equations:

3x + y + z = 0

x – 4y + 3z = 0

2x + 5y – 2z = 0

Any system of equation can be written in matrix form as AX = B

Now finding the Determinant of these set of equations,

= 3(– 4×(– 2) – 3×5) – 1(1×(– 2) – 3×2) + 1(1×5 – 2×(– 4))

= 3(8 – 15) – 1(– 2 – 6) + 1(5 + 8)

= 3×(– 7) –1 × (– 8) + 1×13

= – 21 + 8 + 13

= 0

Since D = 0, so the system of equation has infinite solution.

Now let z = k

⇒ 3x + y = – k

And x – 4y = – 3k

Now using the cramer’s rule

similarly,