Q. 52 A4.7( 3 Votes )

# Solve the following determinant equations:

Let

We need to find the roots of Δ = 0.

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 the term (x + a + b + c) common from C1, we get

Applying R2 R2 – R1, we get

Applying R3 R3 – R1, we get

Expanding the determinant along C1, we have

Δ = (x + a + b + c)(1)[(x)(x) – (0)(0)]

Δ = (x + a + b + c)(x)(x)

Δ = x2(x + a + b + c)

The given equation is Δ = 0.

x2(x + a + b + c) = 0

Case – I:

x2 = 0 x = 0

Case – II:

x + a + b + c = 0 x = –(a + b + c)

Thus, 0 and –(a + b + c) are the roots of the given determinant equation.

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