# 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.

Evaluate the determinant $\left| \begin{matrix}0 & 1 & 3 \\-1 & 2 & 1 \\2 & 1 & 1 \end{matrix} \right|.$

\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}

Evaluate the determinant $\left| \begin{matrix}3 & 4 & -2 \\2 & 4 & -6 \\4 & 2 & 0 \end{matrix} \right|.$

\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}

Evaluate the determinant $\left| \begin{matrix}3 & 1 & 2 \\6 & -5 & 4 \\-9 & 8 & -7 \end{matrix} \right|.$

\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}

Evaluate the determinant $\left| \begin{matrix}2 & 1 & -3 \\1 & 1 & 0 \\-2 & 3 & 4\end{matrix} \right|.$

\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}