Chapter 1: Applications of Matrices and Determinants April 2, 2024 Maven Leave a Comment Welcome to the Chapter 1: Applications of Matrices and Determinants Quiz! This quiz is based on the book back questions. Name Email 1. The rank of the unit matrix of order n is n-1 n n+1 $ n^{2}$ None 2. If A=(1 2 3), then the rank of $ AA^{T} $ is 0 2 3 1 None 3. The rank of m x n matrix whose elements are unity is 0 1 m n None 4. The rank of the diagonal matrix $ \begin{pmatrix} 1 & & & & & \\ & 2 & & & & \\ & & -3 & & & \\ & & & 0 & & \\ & & & & 0 & \\ & & & & & 0 \end{pmatrix} $ 0 2 3 5 None 5. If $ \rho (A)= r $ then which of the following is correct all the minors of order r which does not vanish A has at least one minor of order r which does not vanish A has at least one (r+1) order minor which vanishes All (r+1) and higher order minors should not vanish None 6. If $A = \begin{pmatrix} 2& 0 \\ 0 & 8 \end{pmatrix}then \: \rho (A) \: is$ 0 1 2 n None 7. 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 $ 1 2 3 only real number None 8. 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 $\dfrac{1}{4}$ $\dfrac{1}{5}$ $\dfrac{1}{6}$ $\dfrac{1}{8}$ None 9. The rank of the matrix $ = \begin{pmatrix} 1& 1 &1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{pmatrix}$ is 0 1 2 3 None 10. If $ A =\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix} then \:the\: rank \:of \: AA^{T}$ is 0 1 2 3 None 11. 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 4 0 -4 1 None 12. Rank of a null matrix is 0 -1 $\infty $ 1 None 13. Cramer's rule is applicable only to get a unique solution when $ \Delta_{z} \neq 0$ $ \Delta_{x} \neq 0$ $ \Delta \neq 0$} $ \Delta_{y} \neq 0$ None 14. If $ \dfrac {a_{1}}{x} + \dfrac {b_{1}}{y} = c_{1}, \dfrac {a_{2}}{x} + \dfrac {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 $ \dfrac{\Delta _{2}}{\Delta _{1}}, \dfrac{\Delta _{3}}{\Delta _{1}}$ $ \dfrac{\Delta _{3}}{\Delta _{1}}, \dfrac{\Delta _{2}}{\Delta _{1}}$ $ \dfrac{\Delta _{1}}{\Delta _{2}}, \dfrac{\Delta _{1}}{\Delta _{3}}$ $ \dfrac{-\Delta _{1}}{\Delta _{2}}, \dfrac{-\Delta _{1}}{\Delta _{3}}$ None 15. $ \left | A_{n \times n} \right | =3 and \: \left | adj A \right | =243 $ then the value of n is 4 5 6 7 None 16. If $\rho (A)= \rho (A,B) $ then the system is Consistent and has infinitely many solutions Consistent and has a unique solution Consistent Inconsistent None 17. If $\rho (A) \neq \rho (A,B) $ then the system is Consistent and has infinitely many solutions Consistent and has a unique solution Inconsistent Consistent None 18. In a transition probability matrix, all the entries are greater than or equal to 2 1 0 3 None 19. Which of the following is not an elementary transformation? $R_{i}\leftrightarrow R_{j}$ $R_{i}\rightarrow 2R_{i}+ 2C_{j}$ $R_{i}\rightarrow 2R_{i}- 4R_{j}$ $C_{i}\rightarrow C_{i}+ 5C_{j}$ None 20. The system of equations 4x + 6y = 5, 6x + 9y = 7 has a unique solution no solution infinitely many solutions none of these None 21. If $\rho (A) = \rho (A,B) = $ the number of unknowns, then the system is Consistent and has infinitely many solutions Consistent and has a unique solution Inconsistent Consistent None 22. If the number of variables in a non homogeneous system AX = B is n, then the system possesses a unique solution only when $\rho (A) =\rho (A,B) > n$ $\rho (A) =\rho (A,B) = n$ $\rho (A) =\rho (A,B) < n$ none of these None 23. 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 .2 .3 .4 .7 None 24. If $\left | A \right |\neq 0$ then A is non- singular matrix singular matrix zero matrix none of these None 25. For the system of equations x +2y + 3z = 1, 2x +y + 3z = 2, 5x +5y + 9z = 4, there is only one solution there exists infinitely many solutions there is no solution none of these None Time's up Related Posts:Chapter 1: Matrices and DeterminantsChapter 6: Applications of DifferentiationChapter 4: Cost and Revenue AnalysisChapter 5: Market Structure and Pricing
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