Question 1, Review Exercise

Solutions of Question 1 of Review Exercise of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.

Question 1

Select the best matching option. Chose the correct 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) $n \times n$
    See Answer
    (b): $m \times p$

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) $n \times n$
    See Answer
    (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) $a_{i j}=M_{i j}$
    See Answer
    (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
    See Answer
    (b): multiplication

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) $6$
    See Answer
    ©: $9$

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) $-1$
    See Answer
    (b): $3$

vii. System of homogeneous linear equations has non-trivial solution if:

  • (a) $|A|>0$
  • (b) $|A|<0$
  • (c) $|A|=0$
  • (d) $|A| \neq 0$
    See Answer
    (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
    See Answer
    (c): RankA < 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) $\operatorname{Rank} A=\operatorname{Rank} A_{b}<3$
    See Answer
    (c):$\operatorname{RankA}=\operatorname{RankA}_{b}=3$

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
    See Answer
    (d): all of these

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