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.

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) $n \times n$
  <btn type=“link” collapse=“a1”>See Answer</btn>

  (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) $n \times n$
  <btn type=“link” collapse=“a2”>See Answer</btn>

  (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) $a_{i j}=M_{i j}$
  <btn type=“link” collapse=“a3”>See Answer</btn>

  (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
  <btn type=“link” collapse=“a4”>See Answer</btn>

  (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) $6$
  <btn type=“link” collapse=“a5”>See Answer</btn>

  ©: $9$

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) $-1$
  <btn type=“link” collapse=“a6”>See Answer</btn>

  (b): $3$

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

• (a) $|A|>0$
• (b) $|A|<0$
• (c) $|A|=0$
• (d) $|A| \neq 0$
  <btn type=“link” collapse=“a7”>See Answer</btn>

  (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
  <btn type=“link” collapse=“a8”>See Answer</btn>

  (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) $\operatorname{Rank} A=\operatorname{Rank} A_{b}<3$
  <btn type=“link” collapse=“a9”>See Answer</btn>

  (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
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  (d): all of these

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