Text Book Solutions Exercise1.4 Applications of Matrices and Determinants

Class 12 Samacheer Kalvi students, here are the solutions to Exercise 1.4 in Chapter 1 Applications of Matrices and Determinants. (Business Maths). You can find links to the other exercises also below.

Text Book Solutions Exercise1.4

Choose the correct answer:

1. If A=(1 2 3), then the rank of $AA^{T}$ is
a) 0	b) 2
c) 3	d) 1
2. The rank of m n × matrix whose elements are unity is
a) 0	b) 1
c) m	d) n
3. If $T= \begin{matrix} & \\ A & \\ B& \end{matrix} \begin{pmatrix} A & B\\ .4& .6 \\ .2 & .8 \end{pmatrix}$ is a transition probability matrix, then at equilibrium A is equal to
a) 1/4	b) 1/5
c) 1/6	d) 1/8
4. If $A = \begin{pmatrix} 2& 0 \\ 0 & 8 \end{pmatrix}then \: \rho (A)\: is$
a) 0	b) 1
c) 2	d) n
5. The rank of the matrix $= \begin{pmatrix} 1& 1 &1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{pmatrix}$ is
a) 0	b) 1
c) 2	d) 3
6. The rank of the unit matrix of order n is
a) n-1	b) n
c) n+1	d) $= n^{2}$
7. If $= \rho (A)= r$ then which of the following is correct
(a) all the minors of order r which does not vanish	(b) A has at least one minor of order r which does not vanish
(c) A has at least one (r+1) order minor which vanishes	(d) all (r+1) and higher order minors should not vanish
8. If $A =\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix} then \:the\: rank \:of \: AA^{T}$ is
a) 0	b) 1
c) 2	d) 3
9. If the rank of the matrix If $\begin{pmatrix} \lambda & -1 & 0\\ 0 & \lambda & -1\\ -1 &0 & \lambda \end{pmatrix} is \: 2, \:then \:\lambda \:is$
a) 1	b) 2
c) 3	d) only real number
10. The rank of the diagonal matrix $\begin{pmatrix} 1 & & & & & \\ & 2 & & & & \\ & & -3 & & & \\ & & & 0 & & \\ & & & & 0 & \\ & & & & & 0 \end{pmatrix}$
a) 0	b) 2
c) 3	d) 5
11. If $T= \begin{matrix} & \\ A & \\ B& \end{matrix} \begin{pmatrix} A & B\\ .7& .3 \\ .6 & x \end{pmatrix}$ is a transition probability matrix, then the value of x is
a) 0.2	b) 0.3
c) 0.4	d) 0.7
12. Which of the following is not an elementary transformation?
a) $R_{i}\leftrightarrow R_{j}$	b) $R_{i}\rightarrow 2R_{i}+ 2C_{j}$
c) $R_{i}\rightarrow 2R_{i}- 4R_{j}$	d) $C_{i}\rightarrow C_{i}+ 5C_{j}$
13. If $\rho (A)= \rho (A, B)$ then the system is
a) Consistent and has infinitely many solutions	b) Consistent and has a unique solution
c) Consistent	d) inconsistent
14. If $\rho (A)= \rho (A, B) =$ the number of unknowns, then the system is
a) Consistent and has infinitely many solutions	b) Consistent and has a unique solution
c) inconsistent	d) consistent
15. If $\rho (A) \neq \rho (A, B)$ then the system is
a) Consistent and has infinitely many solutions	b) Consistent and has a unique solution
c) inconsistent	d) consistent
16. In a transition probability matrix, all the entries are greater than or equal to
a) 2	b) 1
c) 0	d) 3
17. If the number of variables in a non homogeneous system AX = B is n, then the system possesses a unique solution only when
a) $\rho (A) =\rho (A, B)> n$	b) $\rho (A) =\rho (A, B)= n$
c) $\rho (A) =\rho (A, B)< n$	d) none of these
18. The system of equations 4x + 6y = 5, 6x + 9y = 7 has
a) a unique solution	b) no solution
c) infinitely many solutions	d) none of these
19. For the system of equations x +2y + 3z = 1, 2x +y + 3z = 2, 5x +5y + 9z = 4,
a) there is only one solution	b) there exists infinitely many solutions
c) there is no solution	d) none of these
20. If $\left \| A \right \|\neq 0$ then A is
a) non- singular matrix	b) singular matrix
c) zero matrix	d) none of these
21. The system of linear equations x + y + z = 2, 2x +y - z = 3, 3x +2y + k = 4, has unique solution, if k is not equal to
a) 4	b) 0
c) -4	d) 1
22. Cramer’s rule is applicable only to get a unique solution when
a) $\Delta_{z} \neq 0$	b) $\Delta_{x} \neq 0$
c) $\Delta \neq 0$ }	d) $\Delta_{y} \neq 0$
23. If $\frac {a_{1}}{x} + \frac {b_{1}}{y} = c_{1}, \frac {a_{2}}{x} + \frac {b_{2}}{y} = c_{2}$
$\Delta _{_{1}=} \begin{vmatrix} a_{1} & b_{1}\\ a_{2} & b_{2} \end{vmatrix}$ , $\Delta _{_{2}=} \begin{vmatrix} b_{1} & c_{1}\\ b_{2} & c_{2} \end{vmatrix}$ $\Delta _{_{3}=} \begin{vmatrix} c_{1} & a_{1}\\ c_{2} & a_{2} \end{vmatrix}$ then (x,y) is
a) $\large \frac{\Delta _{2}}{\Delta _{1}}, \frac{\Delta _{3}}{\Delta _{1}}$	b) $\large \frac{\Delta _{3}}{\Delta _{1}}, \frac{\Delta _{2}}{\Delta _{1}}$
c) $\large \frac{\Delta _{1}}{\Delta _{2}}, \frac{\Delta _{1}}{\Delta _{3}}$	d) $\large \frac{-\Delta _{1}}{\Delta _{2}}, \frac{-\Delta _{1}}{\Delta _{3}}$
24. $\left \| A_{n \times n} \right \| =3 and \: \left \| adj A \right \| =243$ then the value of n is
a) 4	b) 5
c) 6	d) 7
25. Rank of a null matrix is
a) 0	b) -1
c) $\infty$	d) 1