Q. 52 A4.7( 3 Votes )

# Solve the following determinant equations:

Answer :

Let

We need to find the roots of Δ = 0.

Recall that the value of a determinant remains same if we apply the operation R_{i}→ R_{i} + kR_{j} or C_{i}→ C_{i} + kC_{j}.

Applying C_{1}→ C_{1} + C_{2}, we get

Applying C_{1}→ C_{1} + C_{3}, we get

Taking the term (x + a + b + c) common from C_{1}, we get

Applying R_{2}→ R_{2} – R_{1}, we get

Applying R_{3}→ R_{3} – R_{1}, we get

Expanding the determinant along C_{1}, we have

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

⇒ Δ = (x + a + b + c)(x)(x)

∴ Δ = x^{2}(x + a + b + c)

The given equation is Δ = 0.

⇒ x^{2}(x + a + b + c) = 0

__Case – I__:

x^{2} = 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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Solve the matrix equations:

RD Sharma - Volume 1

Using properties of determinants, prove the following:

Mathematics - Board Papers

Using properties of determinants, prove the following:

Mathematics - Board Papers

Using properties of determinants, prove the following:

Mathematics - Board Papers

Solve the matrix equations:

RD Sharma - Volume 1