Matrices & DeterminantsmediumFree

Matrices & Determinants: Vmatrix Vmatrix Vmatrix Vmatrix

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

Question
If Δ1=∣xbbaxbaax∣\Delta_1 = \begin{vmatrix} x & b & b \\ a & x & b \\ a & a & x \end{vmatrix} and Δ2=∣xbax∣\Delta_2 = \begin{vmatrix} x & b \\ a & x \end{vmatrix}, then:
AΔ1=3Δ22\Delta_1 = 3\Delta_2^2
Bddx(Δ1)=3Δ22\dfrac{d}{dx}(\Delta_1) = 3\Delta_2^2
Cddx(Δ1)=3Δ2\dfrac{d}{dx}(\Delta_1) = 3\Delta_2correct
DNone of these
Solution
Step 1: Differentiate Δ1\Delta_1 using the row-differentiation formula
ddx(Δ1)=∣1bb0xb0ax∣+∣x0ba1ba0x∣+∣xb0ax0aa1∣\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 3×33\times3 determinant Each reduces to (x2−ab)=Δ2(x^2 - ab) = \Delta_2:
ddx(Δ1)=Δ2+Δ2+Δ2=3Δ2\frac{d}{dx}(\Delta_1) = \Delta_2 + \Delta_2 + \Delta_2 = 3\Delta_2
 Verification: Δ1=x3−3abx+2ab2\Delta_1 = x^3 - 3abx + 2ab^2, so ddx(Δ1)=3x2−3ab=3(x2−ab)=3Δ2\dfrac{d}{dx}(\Delta_1) = 3x^2-3ab = 3(x^2-ab) = 3\Delta_2. ✓\checkmark Answer: (3)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Matrices & Determinants · easy
If AA is a 3×33 \times 3 matrix and det⁡A=5\det A = 5, then det⁡(Adj(Adj A))\det\big(\text{Adj}(\text{Adj}\,A)\big) equals:
Matrices & Determinants · easy
The equation (12213434k)(xyz)=(000)\begin{pmatrix} 1 & 2 & 2 \\ 1 & 3 & 4 \\ 3 & 4 & k \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} has a solution (x,y,z)(x, y, z) besides (0,0,0)(0, 0, 0). The value of kk equals:
Matrices & Determinants · easy
AA is an involutory matrix given by A=(01−14−343−34)A = \begin{pmatrix}0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\end{pmatrix}, then the inverse of A2\dfrac{A}{2} will be:
Matrices & Determinants · easy
A square matrix PP satisfies P2=I−PP^2 = I - P, where II is the identity matrix. If Pn=5I−8PP^n = 5I - 8P, then nn is:

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath