Exercise 2.2 (Solutions)
The solutions of the Exercise 2.2 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 sum, product and different operations on matrices.
Question 1. Construct a matrix $A=\left[a_{i j}\right]$ of order $2 \times 2$ for which:
(i) $a_{i j}=\dfrac{i+3 j}{2}$
(ii) $a_{i j}=\dfrac{i \times l}{2}$
(iii) $a_{i j}=\dfrac{i}{j}$
(iv) $a_{i j}=\dfrac{2 i-3 j}{3}$
Solution:Question 1
Question 2. Construct a matrix $B=\left[a_{\ell}\right]$ of order $3 \times 3$ for which:
(i) $b_{i j}=\dfrac{t^{2}-j}{3}$
(ii) $b_{i j}=\dfrac{i^{2}-j^{2}}{2 i}$
(iii) $b_{i j}=\dfrac{2}{2 i+j}$
(iv) $b_{i j}=\dfrac{t^{2}+j^{2}}{i+j}$
Solution:Question 2
Question 3. If $A=\left[\begin{array}{ccc}3 & -1 & 2 \\ 0 & 6 & 1 \\ -1 & 0 & -3\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 & 1 & 7 \\ 0 & 2 & -1 \\ -3 & 4 & 2\end{array}\right]$ then find a matrix $C$ such that: $A+B+C=O$
Solution:Question 3
Question 4. (i) Find $A$ if $\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] A\left[\begin{array}{ll}1 & 3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \quad$
(ii) Find $X$ if $\left[\begin{array}{ll}3 & 2 \\ 0 & 1 \\ 2 & 0\end{array}\right] X=\left[\begin{array}{ccc}7 / 2 & 11 & 2 \\ 2 & 4 & 1 \\ 1 & 2 & 0\end{array}\right]$
(iii) If $A=\left[\begin{array}{ll}3 & 7\end{array}\right]$ and $B=\left[\begin{array}{ll}2 & 14\end{array}\right]$ then find a non-zero matrix $C$ such that $A C=B C$.
(iv) $\left[\begin{array}{cc}x y & 4 \\ 0 & x+y\end{array}\right]=\left[\begin{array}{ll}8 & z \\ t & 6\end{array}\right]$ then find the values of $z, t$ and $x^{2}+y^{2}$.
(v) If $A=\left[\begin{array}{ll}3 & 4 \\ 7 & 6\end{array}\right]$ and $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ then find $\alpha$ and $\beta$ so that $A^{2}+\alpha I=\beta A$.
(vi) Find the values of $x$ if $\left[\begin{array}{lll}x & -4 & 2\end{array}\right]\left[\begin{array}{lll}1 & 0 & 3 \\ 0 & 1 & 0 \\ 2 & 0 & 4\end{array}\right]\left[\begin{array}{c}x \\ 1 \\ -1\end{array}\right]=0$.
Solution:Question 4
Question 5. If $X=\left[\begin{array}{lll}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{array}\right]$ then prove that $X^{2}-4 X-5 I=0$.
Solution:Question 5
Question 6. If $A=\left[\begin{array}{cc}2 & 1 \\ 3 & -3\end{array}\right]$ then find $\alpha$ and $\beta$ such that, $A^{2}+\alpha I=\beta A$.
Solution:Question 6
Question 7. If $A=\left[\begin{array}{ll}x & 0 \\ y & 1\end{array}\right]$ then
(i) Prove that for all positive integers $n, A^{n}=\left[\begin{array}{cc}x^{n} & 0 \\ \frac{y\left(x^{n}-1\right)}{x-1} & 1\end{array}\right]$.
(ii) If $A=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ then prove that for all positive integers $n$,
$ A^{n}=\left[\begin{array}{ll}1+2 n & -4 n \\ n & 1-2 n \end{array}\right]$
Solution:Question 7
Question 8. Consider any two particular matrices $A$ and $B$ of your choice of order $2 \times 3$ and $3 \times 2$ respectively and show that $(A B)^{t}=B^{t} A^{t}$.
Solution:Question 8
Question 9. Consider any two particular matrices $A$ and $B$ of your choice of order $3 \times 3$ and show that $(A+B)^{t}=A^{t}+B^{t}$.
Solution:Question 9
Question 10. If $A$ and $B$ are two matrices such that $A B=B$ and $B A=A$. Find $A^{2}+B^{2}$.
Solution:Question 10
Question 11. If $A=\left[a_{i j}\right]$ is a matrix of order $3 \times 3$ and $a_{l j}=i^{2}-j^{2}$. Check whether $A$ is symmetric or skew-symmetric.
Solution:Question 11
Question 12. For any square matrix $A$; prove that $\left(A^{n}\right)^{t}=\left(A^{t}\right)^{n}$.
Solution:Question 12
Question 13. Find the matrices $X$ and $Y$ such that $2 X-Y=\left[\begin{array}{ccc}1 & 6 & -3 \\ 2 & 1 & 7\end{array}\right]$ and $X+3 Y=\left[\begin{array}{ccc}4 & 3 & 2 \\ 1 & -3 & 0\end{array}\right]$.
Solution:Question 13