# 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.

Find the ranks of the matrix. $$\left[ \begin{matrix} 1 & 0 & -2 \\ 2 & 2 & 1 \\ -1 & 2 & 3 \\ \end{matrix} \right]$$

\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 $3$.

Find the ranks of the matrix. $$\left[ \begin{matrix} 3 & 1 & -4 \\ 0 & 2 & 1 \\ 1 & -1 & -2 \\ \end{matrix} \right]$$

\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 $2$.