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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants3 Marks
Question
Evaluate the following determinant: $\begin{vmatrix}\text{a}+\text{ib}&\text{c}+\text{id}\\-\text{c}+\text{id}&\text{a}-\text{ib}\end{vmatrix}$
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Answer

Let $\text{A}=\begin{vmatrix}\text{a}+\text{ib}&\text{c}+\text{id}\\-\text{c}+\text{id}&\text{a}-\text{ib}\end{vmatrix}$
$|\text{A}|=(\text{a}+\text{ib})(\text{a}-\text{ib})-(\text{c}+\text{id})(-\text{c}+\text{id}) ($Taking $(-)$ sign common from $-c +$ id$)$
$=(\text{a}^2+\text{b}^2)+(\text{c}+\text{id})(\text{c}-\text{id}) ($Also $(a + ib)(a - ib) = a^2 + b^2)$
$=\text{a}^2+\text{b}^2+\text{c}^2+\text{d}^2$
Hence, $|\text{A}|=\text{a}^2+\text{b}^2+\text{c}^2+\text{d}^2$

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