====== Question 4, Exercise 2.2 ======
Solutions of Question 4 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 4(i)=====
Evaluate the determinant 
$\left| \begin{matrix}0 & 1 & 3  \\-1 & 2 & 1  \\2 & 1 & 1 \end{matrix} \right|.$
====Solution====
\begin{align}&\left| \begin{matrix}
   0 & 1 & 3  \\
   -1 & 2 & 1  \\
   2 & 1 & 1  \\
\end{matrix} \right| \\
=&0\left( 2-1 \right)-1\left( -1-2 \right)+3\left( -1-4 \right)\\
=&0+3-15=-12 \end{align}

=====Question 4(ii)=====
Evaluate the determinant $\left| \begin{matrix}3 & 4 & -2  \\2 & 4 & -6  \\4 & 2 & 0 \end{matrix} \right|.$
====Solution====
\begin{align}&\left| \begin{matrix}
   3 & 4 & -2  \\
   2 & 4 & -6  \\
   -4 & 2 & 0  \\
\end{matrix} \right|\\
=&3\left( 0+12 \right)-4\left( 0-24 \right)-2\left( 4+16 \right) \\
=&36+96-80=52 \end{align} 

=====Question 4(iii)=====
Evaluate the determinant $\left| \begin{matrix}3 & 1 & 2  \\6 & -5 & 4  \\-9 & 8 & -7 \end{matrix} \right|.$
====Solution====
\begin{align}&\left| \begin{matrix}
   3 & 1 & 2  \\
   6 & -5 & 4  \\
   -9 & 8 & -7  \\
\end{matrix} \right|\\
=&3\left( 35-32 \right)-1\left( -42+36 \right)+2\left( 48-45 \right)\\
=&9+6+6=21 \end{align}

=====Question 4(iv)=====
Evaluate the determinant $\left| \begin{matrix}2 & 1 & -3  \\1 & 1 & 0  \\-2 & 3 & 4\end{matrix} \right|.$
====Solution====
\begin{align}&\left| \begin{matrix}
   2 & 1 & -3  \\
   1 & 1 & 0  \\
   -2 & 3 & 4  \\
\end{matrix} \right|\\
=& 2\left( 4-0 \right)-1\left( 4-0 \right)-3\left( 3+2 \right) \\
=&8-4-15=-11 \end{align}



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