Linear Algebra

1. Let A be an m × n matrix and Ban n × m matrix. It is given that determinant ($$I$$m + AB) = determinant ($$I$$n + BA), where $$I$$k is the k × k identity matrix. Using the above property, the determinant of the matrix given below is \[\left[ {\begin{array}{*{20}{c}} 2&1&1&1 \\ 1&2&1&1 \\ 1&1&2&1 \\ 1&1&1&2 \end{array}} \right]\] A. 2 B. 5 C. 8 D. 16

2
5
8
16

Detailed SolutionLet A be an m × n matrix and Ban n × m matrix. It is given that determinant ($$I$$m + AB) = determinant ($$I$$n + BA), where $$I$$k is the k × k identity matrix. Using the above property, the determinant of the matrix given below is \[\left[ {\begin{array}{*{20}{c}} 2&1&1&1 \\ 1&2&1&1 \\ 1&1&2&1 \\ 1&1&1&2 \end{array}} \right]\] A. 2 B. 5 C. 8 D. 16

3. For the set of equations x1 + 2×2 + x3 + 4×4 = 2 3×1 + 6×2 + 3×3 + 12×4 = 6 the following statement is true: A. Only the trivial solution x1 = x2 = x3 = x4 = 0 exists B. There are no solution C. A unique non-trivial solution exists D. Multiple non-trivial solutions exist

Only the trivial solution x1 = x2 = x3 = x4 = 0 exists
There are no solution
A unique non-trivial solution exists
Multiple non-trivial solutions exist

Detailed SolutionFor the set of equations x1 + 2×2 + x3 + 4×4 = 2 3×1 + 6×2 + 3×3 + 12×4 = 6 the following statement is true: A. Only the trivial solution x1 = x2 = x3 = x4 = 0 exists B. There are no solution C. A unique non-trivial solution exists D. Multiple non-trivial solutions exist

4. If any two columns of a determinant \[{\text{P}} = \left| {\begin{array}{*{20}{c}} 4&7&8 \\ 3&1&5 \\ 9&6&2 \end{array}} \right|\] are interchanged, which one of the following statements regarding the value of the determinant is CORRECT? A. Absolute value remains unchanged but sign will change B. Both absolute value and sign will change C. Absolute value will change but sign will not change D. Both absolute value and sign will remain unchanged

Absolute value remains unchanged but sign will change
Both absolute value and sign will change
Absolute value will change but sign will not change
Both absolute value and sign will remain unchanged

Detailed SolutionIf any two columns of a determinant \[{\text{P}} = \left| {\begin{array}{*{20}{c}} 4&7&8 \\ 3&1&5 \\ 9&6&2 \end{array}} \right|\] are interchanged, which one of the following statements regarding the value of the determinant is CORRECT? A. Absolute value remains unchanged but sign will change B. Both absolute value and sign will change C. Absolute value will change but sign will not change D. Both absolute value and sign will remain unchanged

5. Eigen values of a matrix \[{\text{S}} = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 2&3 \end{array}} \right]\] are 5 and 1. What are the eigen values of the matrix S2 = SS? A. 1 and 25 B. 6 and 4 C. 5 and 1 D. 2 and 10

1 and 25
6 and 4
5 and 1
2 and 10

Detailed SolutionEigen values of a matrix \[{\text{S}} = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 2&3 \end{array}} \right]\] are 5 and 1. What are the eigen values of the matrix S2 = SS? A. 1 and 25 B. 6 and 4 C. 5 and 1 D. 2 and 10

6. The smallest and largest Eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}} 3&{ – 2}&2 \\ 4&{ – 4}&6 \\ 2&{ – 3}&5 \end{array}} \right]\] A. 1.5 and 2.5 B. 0.5 and 2.5 C. 1.0 and 3.0 D. 1.0 and 2.0

1.5 and 2.5
0.5 and 2.5
1.0 and 3.0
1.0 and 2.0

Detailed SolutionThe smallest and largest Eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}} 3&{ – 2}&2 \\ 4&{ – 4}&6 \\ 2&{ – 3}&5 \end{array}} \right]\] A. 1.5 and 2.5 B. 0.5 and 2.5 C. 1.0 and 3.0 D. 1.0 and 2.0

7. Consider the matrix \[\left[ {\begin{array}{*{20}{c}} 5&{ – 1} \\ 4&1 \end{array}} \right]\] . Which one of the following statements is TRUE for the eigen values and eigen vectors of this matrix? A. Eigen value 3 has a multiplicity of 2, and only one independent eigen vector exists B. Eigen value 3 has a multiplicity of 2, and two independent eigen vector exists C. Eigen value 3 has a multiplicity of 2, and no independent eigen vector exists D. Eigen value are 3 and -3, and two independent eigen vectors exist

Eigen value 3 has a multiplicity of 2, and only one independent eigen vector exists
Eigen value 3 has a multiplicity of 2, and two independent eigen vector exists
Eigen value 3 has a multiplicity of 2, and no independent eigen vector exists
Eigen value are 3 and -3, and two independent eigen vectors exist

Detailed SolutionConsider the matrix \[\left[ {\begin{array}{*{20}{c}} 5&{ – 1} \\ 4&1 \end{array}} \right]\] . Which one of the following statements is TRUE for the eigen values and eigen vectors of this matrix? A. Eigen value 3 has a multiplicity of 2, and only one independent eigen vector exists B. Eigen value 3 has a multiplicity of 2, and two independent eigen vector exists C. Eigen value 3 has a multiplicity of 2, and no independent eigen vector exists D. Eigen value are 3 and -3, and two independent eigen vectors exist

8. Let M be a real 4 × 4 matrix. Consider the following statements: S1: M has 4 linearly independent eigenvectors. S2: M has 4 distinct eigenvalues. S3: M is non-singular (invertible). Which one among the following is TRUE? A. S1 implies S2 B. S1 implies S3 C. S2 implies S1 D. S3 implies S2

S1 implies S2
S1 implies S3
S2 implies S1
S3 implies S2

Detailed SolutionLet M be a real 4 × 4 matrix. Consider the following statements: S1: M has 4 linearly independent eigenvectors. S2: M has 4 distinct eigenvalues. S3: M is non-singular (invertible). Which one among the following is TRUE? A. S1 implies S2 B. S1 implies S3 C. S2 implies S1 D. S3 implies S2

10. A is m × n full rank matrix with m > n and $$I$$ is an identity matrix. Let matrix A’ = (ATA)-1AT, Then, which one of the following statement is TRUE? A. AA’ A = A B. (AA’)2 = A C. AA’A = $$I$$ D. AA’A = A’

AA' A = A
(AA')2 = A
AA'A = $$I$$
AA'A = A'

Detailed SolutionA is m × n full rank matrix with m > n and $$I$$ is an identity matrix. Let matrix A’ = (ATA)-1AT, Then, which one of the following statement is TRUE? A. AA’ A = A B. (AA’)2 = A C. AA’A = $$I$$ D. AA’A = A’

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