====== Question 3, Exercise 2.3 ======
Solutions of Question 3 of Exercise 2.3 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(i)=====
Find the ranks of the matrix. 
$$\left[ \begin{matrix}
   1 & 0 & -2  \\
   2 & 2 & 1  \\
   -1 & 2 & 3  \\
\end{matrix} \right]$$ 
====Solution====
\begin{align}&\begin{bmatrix}
1 & 0 & -2  \\
2 & 2 & 1  \\
-1 & 2 & 3 \end{bmatrix}\\
\underset{\sim}{R}& \begin{bmatrix}
1 & 0 & -2  \\
0 & 2 & 5  \\
0 & 2 & 1  \end{bmatrix} \text{ by }R_2-2R_1 \text{ and } R_1-2R_3\\
\underset{\sim}{R}&\begin{bmatrix}
1 & 0 & -2  \\
0 & 0 & 4  \\
0 & 2 & 1 \end{bmatrix}\text{ by }R_2-R_3\end{align}
The last matrix is the echelon form of given matrix having $3$ non-zero rows. 

Hence rank of given matrix is $3$.

=====Question 3(ii)=====
Find the ranks of the matrix.
$$\left[ \begin{matrix}
   3 & 1 & -4  \\
   0 & 2 & 1  \\
   1 & -1 & -2  \\
\end{matrix} \right]$$
====Solution====
\begin{align}&\begin{bmatrix}
3 & 1 & -4  \\ 0 & 2 & 1  \\ 1 & -1 & -2\end{bmatrix}\\ 
\underset{\sim}{R}&\begin{bmatrix}
1 & -1 & -2  \\
0 & 2 & 1  \\
3 & 1 & -4 \end{bmatrix} \text{ by }R_1\leftrightarrow R_3\\
\underset{\sim}{R}&\begin{bmatrix}
1 & -1 & -2  \\
0 & 2 & 1  \\
0 & 4 & 2 \end{bmatrix}\text{ by }R_3-3R_1\\
\underset{\sim}{R}&\begin{bmatrix}
1 & -1 & -2  \\
0 & 2 & 1  \\
0 & 0 & 0 \end{bmatrix} \text{ by }R_3-2R_2\end{align}
The last matrix is the echelon form of given matrix having $2$ non-zero rows. 

Hence rank of the given matrix is $2$.


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