Gujarat 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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