====== Question 11, Exercise 2.2 ======
Solutions of Question 11 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 11(i)=====
Identify singular and non-singular matrices. $\left[ \begin{matrix}7 & 1 & 3  \\6 & 2 & -2  \\5 & 1 & 1\end{matrix} \right]$
====Solution====
Let
$$A=\left[ \begin{matrix}
   7 & 1 & 3  \\
   6 & 2 & -2  \\
   5 & 1 & 1  \\
\end{matrix} \right]$$
$$|A|=7(2+2)-1(6+10)+3(6-10)$$
$$=28-16-12$$
$$|A|=0$$ 
$A$ is singular.

=====Question 11(ii)=====
Identify singular and non-singular matrices. $\left[ \begin{matrix}1 & -1 & 1  \\3 & -2 & 1  \\-2 & -3 & 2 \end{matrix} \right]$
====Solution====
Let
$$A=\left[ \begin{matrix}
   1 & -1 & 1  \\
   3 & -2 & 1  \\
   -2 & -3 & 2  \\
\end{matrix} \right]$$
$$|A|=1(-4+3)+1(6+2)+1(-9-4)$$
$$=-1+8-13$$
$$|A|=-6$$
$A$ is not equal to zero. Then $A$ is non-singular.


=====Question 11(iii)=====
Identify singular and non-singular matrices. $\left[ \begin{matrix}3 & 2 & -3  \\3 & 6 & -3  \\-1 & 0 & 1 \end{matrix} \right]$
====Solution====
Let
$$A=\left[ \begin{matrix}
   3 & 2 & -3  \\
   3 & 6 & -3  \\
   -1 & 0 & 1  \\
\end{matrix} \right]$$
$$|A|=3(6)-2(3-3)-3(6)$$
$$|A|=0$$
$A$ is singular.

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