Review Exercise 2 (Solutions)

The solutions of the Review Exercise 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 MCQs and question all topics included in this chapter.

Question 1. Select the best matching option.
(i) If order of $A$ is $m \times n$ and order of $B$ is $n \times p$ then order of $A B$ is:
(a) $n \times p$
(b) $m \times p$
(c) $p \times m$
(d) $n \times n$

(ii) If $A$ is a row matrix of order $1 \times n$ then order of $A^{t} A$ is:
(a) $1 \times n$
(b) $n \times 1$
(c) $1 \times 1$
(d) $n \times n$

(iii) For an element $a_{i j}$ of a square matrix $A$ :
(a) $a_{i j}=(-1)^{i+j} A_{i j}$
(b) $a_{i j}=(-1)^{i+j} M_{i j}$
(c) $\frac{A_{i j}}{M_{i j}}=(-1)^{i+j}$
(d) $a_{i j}=M_{i j}$

(iv) If $A$ is any matrix then $A$ and $A^{t}$ are always conformable for:
(a) Addition
(b) multiplication
(c) subtraction
(d) all of these

(v) If $A$ is a square matrix of order $3 \times 3$ and $|A|=3$ then value of $|\operatorname{adj} A|$ is:
(a) 3
(b) $1 / 3$
(c) 9
(d) 6

(vi) For the square matrix $A$ of order $3 \times 3$ with $|A|=9 ; A_{21}=2 ; A_{22}=3 ; A_{23}=-1$; $a_{21}=1 ; a_{23}=2$, the value of $a_{22}$ is:
(a) 2
(b) 3
(c) 9
(d) -1

(vii) System of homogeneous linear equations has non-trivial solution if:
(a) $|A|>0$
(b) $|A|<0$
(c) $|A|=0$
(d) $|A| \neq 0$

(viii) For non-homogeneous system of equations; the system is inconsistent if:
(a) $\operatorname{RankA}=\operatorname{Rank} A_{b}$
(b) $\operatorname{RankA} \neq \operatorname{Rank} A_{b}$
(c) RankA < no. of variables
(d) Rank $A_{b}>$ no. of variables

(ix) For a system of non-homogeneous equations with three variables system will have unique solution if:
(a) $\operatorname{RankA}<3$
(b) $\operatorname{Rank} A_{b}<3$
(c) $\operatorname{RankA}=\operatorname{RankA}_{b}=3$
(d) $\operatorname{Rank} A=\operatorname{Rank} A_{b}<3$

(x) A system of non- homogeneous equation having infinite many solutions can be solved by using:
(a) Inversion method
(b) Cramer's rule
(c) Gauss-Jordan method
(d) all of these
Solution:Question 1

Question 2. For the matrix $A=\left[\begin{array}{ccc}1 & 2 & 0 \\ -3 & 4 & 9 \\ 2 & 1 & 6\end{array}\right]$;
find $A_{13}, A_{23}$ and $A_{33}$; hence find $|A|$.

Question 3. Prove that if $A^{-1}=A^{t}$ then $\left|A A^{t}\right|=1$.
Solution:Question 2 & 3

Question 4. Without expanding show that $\left|\begin{array}{ccc}a+1 & l & l \\ l & a+1 & l \\ l & l & a+1\end{array}\right|=(a+1+2 l)(a+1-l)^{2}$.

Question 5. Find the value of $\lambda$ so that the following system has infinite many solutions.
$2 x-3 y+z=1 ; x-2 y+\lambda z=2 ; 3 y+z=-1$
Solution:Question 4 & 5