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CBSE Class 12 Math 2020 Delhi Set 3 Solved Paper

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Question : 1 of 11
Marks: +1, -0
If AA is a skew symmetric matrix of order 3, then the value of ∣A∣\left|A\right| is :____
Solution:  
Since, AA is a skew symmetric matrix
∴AT=−A\therefore A^{T} = -A
⇒∣AT∣=∣−A∣\Rightarrow \left| A^{T} \right| = \left| -A \right|
⇒∣−A∣=(−1)n∣A∣\Rightarrow \left| -A \right| = (-1)^{n} \left| A \right|
where nn is the order.
∴∣−A∣=(−1)3∣A∣\therefore \left| -A \right| = (-1)^{3} \left| A \right|
=−∣A∣= -\left| A \right|
Substituting in (i),
∣AT∣=−∣A∣\left| A^{T} \right| = -\left| A \right|
⇒−∣A∣=∣AT∣\Rightarrow -\left| A \right| = \left| A^{T} \right|
We also know for any matrix.
∣A∣=∣AT∣\left| A \right| = \left| A^{T} \right|
replacing in (ii)
∴∣A∣=−∣A∣\therefore \left| A \right| = -\left| A \right|
⇒2∣A∣=0\Rightarrow 2\left| A \right| = 0
⇒∣A∣=0\Rightarrow \left| A \right| = 0
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