First slide

Introduction to Determinants

Question

Let $D_{1} = |\begin{matrix} a & b & a + b \\ c & d & c + d \\ a & b & a - b \end{matrix}| and D_{2} = |\begin{array}{ccc} a & c & a + c \\ b & d & b + d \\ a & c & a + b + c \end{array}|$ then the value of $|\frac{D_{1}}{D_{2}}|$ , where $b \neq 0 and ad \neq bc$ , is ________ .

Moderate

Solution

Using $C_{3} \to C_{3} - (C_{1} + C_{2}) in D_{1} and D_{2}$ , we have
$\frac{D_{1}}{D_{2}} = \frac{- 2 b (ad - bc)}{b (ad - bc)} = - 2$

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