Question 4, Exercise 2.3

Solutions of Question 4 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 rank of matrix $\begin{bmatrix}2 & 3 & 4 & 5 \\3 & 4 & 5 & 6 \\4 & 5 & 6 & 7 \\9 & 10 & 11 & 12\end{bmatrix}$

$$\begin{align}&\begin{bmatrix} 2 & 3 & 4 & 5 \\ 3 & 4 & 5 & 6 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix}\\ \underset{\sim}{R}&\begin{bmatrix} 2 & 3 & 4 & 5 \\ 1 & 1 & 1 & 1 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix} \text{by}R_2-R_1\\ \underset{\sim}{R}&\begin{bmatrix} 1 & 1 & 1 & 1 \\ 2 & 3 & 4 & 5 \\ 4 & 5 & 6 & 7 \\ 9 & 10 & 11 & 12 \end{bmatrix} \text{by}R_1\leftrightarrow R_2\\ \underset{\sim}{R}&\begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 \end{bmatrix} \text{by}R_2-2R_1\text{,}R_3-4R_1 \text{and} R_4-9R_1\\ \underset{\sim}{R}&\begin{bmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \text{by}R_4-R_2 \text{and} R_3-R_2\end{align}$$ 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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