First slide

Introduction to Determinants

Question

If $Δ_{1} = |\begin{matrix} x & b & b \\ a & x & b \\ a & a & x \end{matrix}| and Δ_{2} = |\begin{matrix} x & b \\ a & x \end{matrix}|$ are the given determinants, then

Moderate

A

$Δ_{1} = 3 {(Δ_{2})}^{2}$
B

$\frac{d}{dx} (Δ_{1}) = 3 Δ_{2}$
C

$\frac{d}{dx} (Δ_{1}) = 3 {(Δ_{2})}^{2}$
D

$Δ_{1} = 3 Δ_{2}^{3 / 2}$

Solution

$Δ_{1} = x (x^{2} - ab) - b (ax - ab) + b (a^{2} - ax) = x^{3} - 3 abx + {ab}^{2} + a^{2} b$
$\frac{d}{dx} (Δ_{1}) = 3 x^{2} - 3 ab = 3 (x^{2} - ab) = 3 Δ_{2}$

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