Exercise 2.5 (Solutions)

The solutions of the Exercise 2.5 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to solving system of the equation of three variables by using matrices.

Question. 1 First reduce each of the following matrices into echelon form then into reduced echelon form.
(i) $\left[\begin{array}{ccc}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{array}\right]$

(ii) $\left[\begin{array}{ll}2 & 1 \\ 3 & 2 \\ 1 & 9\end{array}\right]$

(iii) $\left[\begin{array}{ccc}2 & -1 & 0 \\ 4 & 7 & 8 \\ -3 & 1 & 3\end{array}\right]$

(iv) $\left[\begin{array}{ccc}2 & -4 & 3 \\ 4 & 1 & 8 \\ 7 & 3 & 0\end{array}\right]$

(v) $\left[\begin{array}{lll}3 & 1 & 2 \\ 2 & 9 & 8\end{array}\right]$

(vi) $\left[\begin{array}{lll}0 & 2 & 4 \\ 0 & 3 & 6 \\ 0 & 1 & 2\end{array}\right]$ Solution:Question 1

Question. 2 Find the rank of each of the following matrices.
(i) $\left[\begin{array}{ccc}5 & 9 & 3 \\ 3 & -5 & 6 \\ 2 & 10 & 6\end{array}\right]$

(ii) $\left[\begin{array}{ccc}-1 & -2 & 3 \\ -1 & 2 & -1 \\ -5 & 2 & 3\end{array}\right]$

(iii) $\left[\begin{array}{ccc}3 & 2 & 4 \\ 2 & 1 & 6 \\ 4 & -1 & 0\end{array}\right]$

(iv) $\left[\begin{array}{ll}1 & 3 \\ 2 & 9 \\ 1 & 6\end{array}\right]$
Solution:Question 2

Question. 3 With the help of row operations, find the inverse of the following matrices if it exists.
Also verify your answer by showing that $A A^{-1}=A^{-1} A=I$.
(i) $\left[\begin{array}{ccc}0 & -1 & -1 \\ -1 & 3 & 0 \\ 1 & -1 & 4\end{array}\right]$

(ii) $\left[\begin{array}{ccc}1 & 2 & 5 \\ -3 & 0 & 1 \\ 4 & 2 & 5\end{array}\right]$

(iii) $\left[\begin{array}{ccc}-5 & 2 & 3 \\ -1 & -2 & 3 \\ 1 & -2 & 3\end{array}\right]$

(iv) $\left[\begin{array}{ccc}0 & 1 & 3 \\ 3 & 2 & 4 \\ 6 & -1 & 2\end{array}\right]$
Solution:Question 3