Q. 39

If a, b and c are all non-zero and , then prove that .(CBSE 2016)

Answer :

Let


Given that Δ = 0.


We can write the determinant Δ as



Taking a, b and c common from C1, C2 and C3, we get



Recall that the value of a determinant remains same if we apply the operation Ri Ri + kRj or Ci Ci + kCj.


Applying C1 C1 + C2, we get




Applying C1 C1 + C3, we get




Taking common from C1, we get



Applying R2 R2 – R1, we get




Applying R3 R3 – R1, we get




Expanding the determinant along C1, we have




We have Δ = 0.



It is given that a, b and c are all non-zero.



Thus, when and a, b, c are all non-zero.

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