To Find The number of solutions of the following system of linear homogeneous equations x−y+z=0 ...(i) x+2y−z=0 ...(ii) 2x+y+3z=0 ...(iii) We can write above equations in matrix form i.e. [
1
−1
1
1
2
−1
2
1
3
][
x
y
z
]=0 or AX=0, where A=[
1
−1
1
1
2
−1
2
1
3
] Since, we know that, if |A|=0, then only non-trivial solution exist. Consider, |A|=[
1
−1
1
1
2
−1
2
1
3
] ∴ |A|=1[2×3−(−1)]+1[3+2]+1[1−4] =7+5−3=9≠0 ∵ |A|≠0 Hence, there is no non-trivial solutions. ⇒ There will be only l trivial solution.