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Matrices & Determinants: Vmatrix Vmatrix Vmatrix Vmatrix

JEE Maths question with a full step-by-step solution.

Question

\Delta_1 = \begin{vmatrix} x & b & b \\ a & x & b \\ a & a & x \end{vmatrix}

and

\Delta_2 = \begin{vmatrix} x & b \\ a & x \end{vmatrix}

, then:

\Delta_1 = 3\Delta_2^2

\dfrac{d}{dx}(\Delta_1) = 3\Delta_2^2

\dfrac{d}{dx}(\Delta_1) = 3\Delta_2

correct

DNone of these

Solution

Step 1: Differentiate

\Delta_1

using the row-differentiation formula

\frac{d}{dx}(\Delta_1) = \begin{vmatrix} 1 & b & b \\ 0 & x & b \\ 0 & a & x \end{vmatrix} + \begin{vmatrix} x & 0 & b \\ a & 1 & b \\ a & 0 & x \end{vmatrix} + \begin{vmatrix} x & b & 0 \\ a & x & 0 \\ a & a & 1 \end{vmatrix}

Step 2: Evaluate each

3\times3

determinant Each reduces to

(x^2 - ab) = \Delta_2

\frac{d}{dx}(\Delta_1) = \Delta_2 + \Delta_2 + \Delta_2 = 3\Delta_2

Verification:

\Delta_1 = x^3 - 3abx + 2ab^2

, so

\dfrac{d}{dx}(\Delta_1) = 3x^2-3ab = 3(x^2-ab) = 3\Delta_2

\checkmark

Answer: (3)

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