Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants2 Marks
Question
If $A$ is a square matrix satisfying $A^T A = l,$ 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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