Linear Algebra

21. Let the eigen values of a 2 × 2 matrix A be 1, -2 with eigen vectors x1 and x2 respectively. Then the eigen values and eigen vectors of the matrix A2 – 3A + 4$$I$$ would, respectively, be A. 2, 14; x1, x2 B. 2, 14; x1 + x2, x1 – x2 C. 2, 0; x1, x2 D. 2, 0; x1 + x2, x1 – x2

2, 14; x1, x2
2, 14; x1 + x2, x1 - x2
2, 0; x1, x2
2, 0; x1 + x2, x1 - x2

Detailed SolutionLet the eigen values of a 2 × 2 matrix A be 1, -2 with eigen vectors x1 and x2 respectively. Then the eigen values and eigen vectors of the matrix A2 – 3A + 4$$I$$ would, respectively, be A. 2, 14; x1, x2 B. 2, 14; x1 + x2, x1 – x2 C. 2, 0; x1, x2 D. 2, 0; x1 + x2, x1 – x2

22. Consider the system of simultaneous equations x + 2y + z = 6 2x + y + 2z = 6 x + y + z = 5 This system has A. unique solution B. infinite number of solutions C. no solution D. exactly two solutions

unique solution
infinite number of solutions
no solution
exactly two solutions

Detailed SolutionConsider the system of simultaneous equations x + 2y + z = 6 2x + y + 2z = 6 x + y + z = 5 This system has A. unique solution B. infinite number of solutions C. no solution D. exactly two solutions

23. The lowest Eigen value of the 2 × 2 matrix \[\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&3 \end{array}} \right]\] A. 1 B. 2 C. 3 D. 5

1
2
3
5

Detailed SolutionThe lowest Eigen value of the 2 × 2 matrix \[\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&3 \end{array}} \right]\] A. 1 B. 2 C. 3 D. 5

24. Which one of the following statements is TRUE about every n × n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive

If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative
If the trace of the matrix is positive, all its eigen values are positive
If the determinant of the matrix is positive, all its eigen values are positive
If the product of the trace and determinant of the matrix is positive, all its eigen values are positive

Detailed SolutionWhich one of the following statements is TRUE about every n × n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive

25. One of the eigen vectors of the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&2 \\ 1&3 \end{array}} \right]\] is A. \[\left\{ {\begin{array}{*{20}{c}} 2 \\ { – 1} \end{array}} \right\}\] B. \[\left\{ {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right\}\] C. \[\left\{ {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right\}\] D. \[\left\{ {\begin{array}{*{20}{c}} 1 \\ { – 1} \end{array}} \right\}\]

”[left{
” option2=”\[\left\{ {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right\}\]” option3=”\[\left\{ {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right\}\]” option4=”\[\left\{ {\begin{array}{*{20}{c}} 1 \\ { – 1} \end{array}} \right\}\]” correct=”option3″]

Detailed SolutionOne of the eigen vectors of the matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 2&2 \\ 1&3 \end{array}} \right]\] is A. \[\left\{ {\begin{array}{*{20}{c}} 2 \\ { – 1} \end{array}} \right\}\] B. \[\left\{ {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right\}\] C. \[\left\{ {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right\}\] D. \[\left\{ {\begin{array}{*{20}{c}} 1 \\ { – 1} \end{array}} \right\}\]

26. Consider the 5 × 5 matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&2&3&4&5 \\ 5&1&2&3&4 \\ 4&5&1&2&3 \\ 3&4&5&1&2 \\ 2&3&4&5&1 \end{array}} \right]\] It is given that A has only one real eigen value. Then the real eigen value of A is A. -2.5 B. 0 C. 15 D. 25

-2.5
0
15
25

Detailed SolutionConsider the 5 × 5 matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&2&3&4&5 \\ 5&1&2&3&4 \\ 4&5&1&2&3 \\ 3&4&5&1&2 \\ 2&3&4&5&1 \end{array}} \right]\] It is given that A has only one real eigen value. Then the real eigen value of A is A. -2.5 B. 0 C. 15 D. 25

27. The rank of the matrix \[{\text{M}} = \left[ {\begin{array}{*{20}{c}} 5&{10}&{10} \\ 1&0&2 \\ 3&6&6 \end{array}} \right]\] is A. 0 B. 1 C. 2 D. 3

0
1
2
3

Detailed SolutionThe rank of the matrix \[{\text{M}} = \left[ {\begin{array}{*{20}{c}} 5&{10}&{10} \\ 1&0&2 \\ 3&6&6 \end{array}} \right]\] is A. 0 B. 1 C. 2 D. 3

28. The eigen values of matrix \[\left[ {\begin{array}{*{20}{c}} 9&5 \\ 5&8 \end{array}} \right]\] are A. -2.42 and 6.86 B. 3.48 and 13.53 C. 4.70 and 6.86 D. 6.86 and 9.50

-2.42 and 6.86
3.48 and 13.53
4.70 and 6.86
6.86 and 9.50

Detailed SolutionThe eigen values of matrix \[\left[ {\begin{array}{*{20}{c}} 9&5 \\ 5&8 \end{array}} \right]\] are A. -2.42 and 6.86 B. 3.48 and 13.53 C. 4.70 and 6.86 D. 6.86 and 9.50

29. For the matrix \[\left[ {\begin{array}{*{20}{c}} 4&2 \\ 2&4 \end{array}} \right]\] the eigen value corresponding to the eigen vector \[\left[ {\begin{array}{*{20}{c}} {101} \\ {101} \end{array}} \right]\] is A. 2 B. 4 C. 6 D. 8

2
4
6
8

Detailed SolutionFor the matrix \[\left[ {\begin{array}{*{20}{c}} 4&2 \\ 2&4 \end{array}} \right]\] the eigen value corresponding to the eigen vector \[\left[ {\begin{array}{*{20}{c}} {101} \\ {101} \end{array}} \right]\] is A. 2 B. 4 C. 6 D. 8

30. One of the eigen vectors of matrix is \[\left[ {\begin{array}{*{20}{c}} { – 5}&2 \\ { – 9}&6 \end{array}} \right]\] is A. \[\left\{ {\begin{array}{*{20}{c}} { – 1} \\ 1 \end{array}} \right\}\] B. \[\left\{ {\begin{array}{*{20}{c}} { – 2} \\ 9 \end{array}} \right\}\] C. \[\left\{ {\begin{array}{*{20}{c}} 2 \\ { – 1} \end{array}} \right\}\] D. \[\left\{ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right\}\]

”[left{
” option2=”\[\left\{ {\begin{array}{*{20}{c}} { – 2} \\ 9 \end{array}} \right\}\]” option3=”\[\left\{ {\begin{array}{*{20}{c}} 2 \\ { – 1} \end{array}} \right\}\]” option4=”\[\left\{ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right\}\]” correct=”option3″]

Detailed SolutionOne of the eigen vectors of matrix is \[\left[ {\begin{array}{*{20}{c}} { – 5}&2 \\ { – 9}&6 \end{array}} \right]\] is A. \[\left\{ {\begin{array}{*{20}{c}} { – 1} \\ 1 \end{array}} \right\}\] B. \[\left\{ {\begin{array}{*{20}{c}} { – 2} \\ 9 \end{array}} \right\}\] C. \[\left\{ {\begin{array}{*{20}{c}} 2 \\ { – 1} \end{array}} \right\}\] D. \[\left\{ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right\}\]

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