Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS2 Marks
Question
If A is a square matrix satisfying AT A = l, write the value of |A|.
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Answer
Let $|\text{A}|=|\text{A}|^{\text{T}} $ [By property of determinants] Given, $\text{A}^{\text{T}}\text{A}=\text{I}$ $\Rightarrow|\text{A}^{\text{T}}\text{A}|=1$ Then, $|\text{A}^{\text{T}}\text{A}|=|\text{A}^{\text{T}}||\text{A}|$ [Since the determinants are of the same order] $\Rightarrow|\text{A}^{\text{T}}||\text{A}|=1$ $\Rightarrow|\text{A}|=\frac{1}{|\text{A}^{\text{T}}|}$ $\Rightarrow|\text{A}|=\frac{1}{|\text{A}|}$$\big[\therefore|\text{A}|=|\text{A}^{\text{T}}|\big]$ $\Rightarrow|\text{A}|^2=1$ $\Rightarrow|\text{A}|=\pm1$
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