====== Question 3, Exercise 2.2 ======
Solutions of Question 3 of Exercise 2.2 of Unit 02: Matrices and Determinants. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.
=====Question 3=====
Let $A$ be square matrix of order $3,$ then verify that $|A^t|=|A|$.
====Solution====
Let
$$A=\begin{bmatrix}
   a_{11} & a_{12} & a_{13}  \\
   a_{21} & a_{22} & a_{23}  \\
   a_{31} & a_{32} & a_{33}  \\
\end{bmatrix}$$
Then
\begin{align}|A|&=a_{11} \left( a_{22} a_{33}-a_{23} a_{32} \right)-a_{12}\left( a_{21}a_{33}-a_{23}a_{31} \right)+a_{13}\left( a_{21}a_{32}-a_{22}a_{31} \right)\\
&=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}\\
&=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}-a_{13}a_{22}a_{31} \ldots (1)
\end{align}
Now
$$
{{A}^{t}}=\left[ \begin{matrix}
   {{a}_{11}} & {{a}_{21}} & {{a}_{31}}  \\
   {{a}_{12}} & {{a}_{22}} & {{a}_{32}}  \\
   {{a}_{13}} & {{a}_{23}} & {{a}_{33}}  \\
\end{matrix} \right]\\
$$
Then
\begin{align}
|A^t|&=a_{11}\left( a_{22}a_{33}-a_{32}a_{23} \right)-a_{21}\left( a_{12}a_{33}-a_{32}a_{13} \right)+a_{31}\left( a_{12}a_{23}-a_{22}a_{13} \right)\\
&=a_{11}a_{22}a_{33}-a_{11}a_{32}a_{23}-a_{21}a_{12}a_{33}+a_{21}a_{32}a_{13}+a_{31}a_{12}a_{23}-a_{31}a_{22}a_{13}\\
&=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}-a_{13}a_{22}a_{31} \ldots (2)
\end{align}
Now comparing (1) and (2), we have
$$|A|=|{{A}^{t}}|.$$


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