Linear Algebra

21. Let A be a 4 × 3 real matrix with rank 2. Which one of the following statement is TRUE? A. Rank of ATA is less than 2 B. Rank of ATA is equal to 2 C. Rank of ATA is greater than 2 D. Rank of ATA can be any number between 1 and 3

Rank of ATA is less than 2
Rank of ATA is equal to 2
Rank of ATA is greater than 2
Rank of ATA can be any number between 1 and 3

Detailed SolutionLet A be a 4 × 3 real matrix with rank 2. Which one of the following statement is TRUE? A. Rank of ATA is less than 2 B. Rank of ATA is equal to 2 C. Rank of ATA is greater than 2 D. Rank of ATA can be any number between 1 and 3

22. The inverse of matrix \[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}} \right]\] is A. \[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}} \right]\] B. \[\left[ {\begin{array}{*{20}{c}} 0&{ – 1}&0 \\ { – 1}&0&0 \\ 0&0&{ – 1} \end{array}} \right]\] C. \[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}} \right]\] D. \[\left[ {\begin{array}{*{20}{c}} 0&{ – 1}&0 \\ 0&0&{ – 1} \\ { – 1}&0&0 \end{array}} \right]\]

”[left[
\]” option2=”\[\left[ {\begin{array}{*{20}{c}} 0&{ – 1}&0 \\ { – 1}&0&0 \\ 0&0&{ – 1} \end{array}} \right]\]” option3=”\[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}} \right]\]” option4=”\[\left[ {\begin{array}{*{20}{c}} 0&{ – 1}&0 \\ 0&0&{ – 1} \\ { – 1}&0&0 \end{array}} \right]\]” correct=”option3″]

Detailed SolutionThe inverse of matrix \[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}} \right]\] is A. \[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 1&0&0 \\ 0&0&1 \end{array}} \right]\] B. \[\left[ {\begin{array}{*{20}{c}} 0&{ – 1}&0 \\ { – 1}&0&0 \\ 0&0&{ – 1} \end{array}} \right]\] C. \[\left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}} \right]\] D. \[\left[ {\begin{array}{*{20}{c}} 0&{ – 1}&0 \\ 0&0&{ – 1} \\ { – 1}&0&0 \end{array}} \right]\]

25. Consider the systems, each consisting of m linear equations in n variables. I. If m < n, then all such systems have a solution. II. If m > n, then none of these systems has a solution. III. If m = n, then there exists a system which has a solution. Which one of the following is CORRECT? A. I, II and III are true B. Only II and III are true C. Only III is true D. None of them is true

I, II and III are true
Only II and III are true
Only III is true
None of them is true

Detailed SolutionConsider the systems, each consisting of m linear equations in n variables. I. If m < n, then all such systems have a solution. II. If m > n, then none of these systems has a solution. III. If m = n, then there exists a system which has a solution. Which one of the following is CORRECT? A. I, II and III are true B. Only II and III are true C. Only III is true D. None of them is true

26. Consider the following system of linear equations \[\left[ {\begin{array}{*{20}{c}} 2&1&{ – 4} \\ 4&3&{ – 12} \\ 1&2&{ – 8} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\text{x}} \\ {\text{y}} \\ {\text{z}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \alpha \\ 5 \\ 7 \end{array}} \right]\] Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of \[\alpha \], does this system of equations have infinitely many solutions? A. 0 B. 1 C. 2 D. infinitely many

0
1
2
infinitely many

Detailed SolutionConsider the following system of linear equations \[\left[ {\begin{array}{*{20}{c}} 2&1&{ – 4} \\ 4&3&{ – 12} \\ 1&2&{ – 8} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\text{x}} \\ {\text{y}} \\ {\text{z}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} \alpha \\ 5 \\ 7 \end{array}} \right]\] Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of \[\alpha \], does this system of equations have infinitely many solutions? A. 0 B. 1 C. 2 D. infinitely many

27. x + 2y + z = 4 2x + y + 2z = 5 x – y + z = 1 The system of algebraic given below has A. A unique solution of x = 1, y = 1 and z = 1 B. only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0) C. infinite number of solutions D. no feasible solution

A unique solution of x = 1, y = 1 and z = 1
only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0)
infinite number of solutions
no feasible solution

Detailed Solutionx + 2y + z = 4 2x + y + 2z = 5 x – y + z = 1 The system of algebraic given below has A. A unique solution of x = 1, y = 1 and z = 1 B. only the two solutions of (x = 1, y = 1, z = 1) and (x = 2, y = 1, z = 0) C. infinite number of solutions D. no feasible solution

28. Which one of the following statements is true for all real symmetric matrices? A. All the eigen values are real B. All the eigen values are positive C. All the eigen values are distinct D. Sum of all the eigen values is zero

All the eigen values are real
All the eigen values are positive
All the eigen values are distinct
Sum of all the eigen values is zero

Detailed SolutionWhich one of the following statements is true for all real symmetric matrices? A. All the eigen values are real B. All the eigen values are positive C. All the eigen values are distinct D. Sum of all the eigen values is zero

29. Given an orthogonal matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&1&1&1 \\ 1&1&{ – 1}&{ – 1} \\ 1&{ – 1}&0&0 \\ 0&0&1&{ – 1} \end{array}} \right],\,{\left[ {{\text{A}}{{\text{A}}^{\text{T}}}} \right]^{ – 1}}\,{\text{is}}\] A. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{4}}&0&0&0 \\ 0&{\frac{1}{4}}&0&0 \\ 0&0&{\frac{1}{2}}&0 \\ 0&0&0&{\frac{1}{2}} \end{array}} \right]\] B. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{2}}&0&0&0 \\ 0&{\frac{1}{2}}&0&0 \\ 0&0&{\frac{1}{2}}&0 \\ 0&0&0&{\frac{1}{2}} \end{array}} \right]\] C. \[\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right]\] D. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{4}}&0&0&0 \\ 0&{\frac{1}{4}}&0&0 \\ 0&0&{\frac{1}{4}}&0 \\ 0&0&0&{\frac{1}{4}} \end{array}} \right]\]

”[left[
\]” option2=”\[\left[ {\begin{array}{*{20}{c}} {\frac{1}{2}}&0&0&0 \\ 0&{\frac{1}{2}}&0&0 \\ 0&0&{\frac{1}{2}}&0 \\ 0&0&0&{\frac{1}{2}} \end{array}} \right]\]” option3=”\[\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right]\]” option4=”\[\left[ {\begin{array}{*{20}{c}} {\frac{1}{4}}&0&0&0 \\ 0&{\frac{1}{4}}&0&0 \\ 0&0&{\frac{1}{4}}&0 \\ 0&0&0&{\frac{1}{4}} \end{array}} \right]\]” correct=”option1″]

Detailed SolutionGiven an orthogonal matrix \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&1&1&1 \\ 1&1&{ – 1}&{ – 1} \\ 1&{ – 1}&0&0 \\ 0&0&1&{ – 1} \end{array}} \right],\,{\left[ {{\text{A}}{{\text{A}}^{\text{T}}}} \right]^{ – 1}}\,{\text{is}}\] A. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{4}}&0&0&0 \\ 0&{\frac{1}{4}}&0&0 \\ 0&0&{\frac{1}{2}}&0 \\ 0&0&0&{\frac{1}{2}} \end{array}} \right]\] B. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{2}}&0&0&0 \\ 0&{\frac{1}{2}}&0&0 \\ 0&0&{\frac{1}{2}}&0 \\ 0&0&0&{\frac{1}{2}} \end{array}} \right]\] C. \[\left[ {\begin{array}{*{20}{c}} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right]\] D. \[\left[ {\begin{array}{*{20}{c}} {\frac{1}{4}}&0&0&0 \\ 0&{\frac{1}{4}}&0&0 \\ 0&0&{\frac{1}{4}}&0 \\ 0&0&0&{\frac{1}{4}} \end{array}} \right]\]

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