Q. 52 F5.0( 1 Vote )

# 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 R_{2}→ R_{2} – R_{1}, we get

Applying R_{3}→ R_{3} – R_{1}, we get

Taking (b – x) and (c – x) common from R_{2} and R_{3}, we get

Expanding the determinant along C_{1}, we have

Δ = (b – x)(c – x)(1)[(1)(c^{2} + cx + x^{2}) – (1)(b^{2} + bx + x^{2})]

⇒ Δ = (b – x)(c – x)[c^{2} + cx + x^{2} – b^{2} – bx – x^{2}]

⇒ Δ = (b – x)(c – x)[c^{2} – b^{2} + cx – bx]

⇒ Δ = (b – x)(c – x)[(c – b)(c + b) + (c – b)x]

∴ Δ = (b – x)(c – x)(c – b)(c + b + x)

The given equation is Δ = 0.

⇒ (b – x)(c – x)(c – b)(c + b + x) = 0

However, b ≠ c according to the given condition.

⇒ (b – x)(c – x)(c + b + x) = 0

__Case – I__:

b – x = 0 ⇒ x = b

__Case – II__:

c – x = 0 ⇒ x = c

__Case – III__:

c + b + x = 0 ⇒ x = –(b + c)

Thus, b, c and –(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