Virtual Labs

1. The identity matrix of order 3 is:

          1  0  0
          0  1  0
          0  0  1

Explanation

          0  1  0
          1  0  1
          0  1  0

Explanation

          1  1  1
          0  0  0
          0  0  0

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          0  0  0
          0  0  0
          0  0  0

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2. If A =

          2  3
          1  4

and B =

          1  0
          2  1

, then A + B equals:

          3  3
          3  5

Explanation

          1  3
          1  5

Explanation

          4  3
          2  3

Explanation

          3  2
          2  4

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3. If A =

          a  b
          c  d

, then det(A) equals:

a: a + d Explanation

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b: ad - bc Explanation

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c: ab - cd Explanation

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d: a - d Explanation

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4. For vectors u and v of length n, the outer product u vᵀ has size:

a: n × n Explanation

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b: 1 × n Explanation

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c: n × 1 Explanation

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d: 1 × 1 Explanation

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5. The transpose of A =

          3  8  4
          -6  -1  -4
          7  5  -2

is:

          3  -6  7
          8  -1  5
          4  -4  -2

Explanation

          3  8  4
          -6  -1  -4
          7  5  -2

Explanation

          3  7  -6
          8  -1  -4
          4  5  -2

Explanation

          7  5  -2
          3  8  4
          -6  -1  -4

Explanation

6. A matrix that is NOT invertible is called:

a: Singular matrix Explanation

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b: Poor matrix Explanation

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c: Linearly independent matrix Explanation

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d: Orthogonal matrix Explanation

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7. The rank of a matrix equals the:

a: Maximum number of linearly independent columns Explanation

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b: Minimum number of independent rows Explanation

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c: Total number of elements Explanation

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d: Rows minus columns Explanation

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8. Which condition implies an n×n matrix is invertible?

a: All of these Explanation

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b: Rank = n Explanation

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c: det(A) ≠ 0 Explanation

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d: Ax = 0 implies x = 0 Explanation

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9. Vectors u =

          2  4  2

and v =

          1  2  1

are:

a: Linearly independent but not orthogonal Explanation

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b: Orthogonal but not independent Explanation

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c: Neither Explanation

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d: Both Explanation

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10. Vectors a₁,...,aₙ are linearly independent when:

a: If all αᵢ = 0 then Σ αᵢ aᵢ = 0 Explanation

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b: There exist non-zero αᵢ making Σ αᵢ aᵢ = 0 Explanation

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c: There exist non-zero αᵢ making Σ αᵢ aᵢ ≠ 0 Explanation

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d: Σ αᵢ aᵢ = 0 implies αᵢ = 0 Explanation

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11. Vectors are linearly dependent when:

a: There exist non-zero αᵢ such that Σ αᵢ aᵢ = 0 Explanation

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b: Σ αᵢ aᵢ = 0 iff αᵢ = 0 Explanation

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c: Zero vector implies Σ = 0 Explanation

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d: Non-zero entries imply Σ ≠ 0 Explanation

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12. Eigenvalues of a Hermitian matrix are:

a: Real but not necessarily positive Explanation

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b: Always positive Explanation

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c: Purely imaginary Explanation

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d: Complex Explanation

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13. If matrix A =

          1  2
          3  4

which of the following statements is true about its eigenvalues?

a: They are both real and distinct. Explanation

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b: They are complex conjugates. Explanation

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c: They are equal. Explanation

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d: They are purely imaginary. Explanation

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14. Let A =

          2  0
          0  3

Which matrix B commutes with A (i.e., AB = BA)?

          0  1
          1  0

Explanation

          1  1
          0  1

Explanation

          a  0
          0  b

Explanation

          0  2
          2  0

Explanation

15. Consider matrix A =

          0  1
          -2  -3

What is the trace and determinant of A?

a: Trace = -3, Determinant = -2 Explanation

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b: Trace = -3, Determinant = 2 Explanation

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c: Trace = -2, Determinant = 3 Explanation

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d: Trace = 0, Determinant = -3 Explanation

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16. For a 3×3 matrix A, if its rank is 2, what can be inferred?

a: It has 1 linearly dependent row. Explanation

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b: It is invertible. Explanation

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c: Its determinant is non-zero. Explanation

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d: All rows are linearly independent. Explanation

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17. If a matrix A is orthogonal, which of the following is always true?

a: A^T = A Explanation

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b: A^T = A⁻¹ Explanation

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c: det(A) = 0 Explanation

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d: All entries are either 0 or 1 Explanation

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18. If matrix A has eigenvalues 1, 2, and 3, what is the determinant of A?

a: 0 Explanation

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b: 6 Explanation

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c: 5 Explanation

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d: Cannot be determined Explanation

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19. If A is a square matrix such that A² = I, what are the possible eigenvalues of A?

a: 1 and -1 Explanation

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b: 0 and 1 Explanation

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c: 2 and -2 Explanation

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d: Only 1 Explanation

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20. Which of the following matrices is not invertible?

          1  2
          2  4

Explanation

          2  0
          0  3

Explanation

          0  1
          1  0

Explanation

          5  1
          2  3

Explanation

21. If A is a 3×3 matrix and det(A) = 5, what is det(2A)?

a: 10 Explanation

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b: 20 Explanation

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c: 40 Explanation

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d: 8 Explanation

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22. Which statement is true for any symmetric real matrix?

a: It is always invertible. Explanation

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b: It always has real eigenvalues. Explanation

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c: Its determinant is always positive. Explanation

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d: It must be diagonal. Explanation

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23. If A is a 2×2 invertible matrix, which of the following must also be invertible?

a: 2A Explanation

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b: A² Explanation

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c: A⁻¹ Explanation

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d: All of the above Explanation

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24. What is the rank of matrix

          1  2
          2  4

a: 0 Explanation

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b: 1 Explanation

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c: 2 Explanation

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d: 3 Explanation

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25. Which of the following matrices is symmetric?

          1  2
          3  4

Explanation

          2  -1
          -1  2

Explanation

          0  1
          1  0

Explanation

          1  0
          0  -1

Explanation

26. What is the trace of matrix

          2  1
          3  4

a: 5 Explanation

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b: 6 Explanation

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c: 2 Explanation

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d: 7 Explanation

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27. Which transformation preserves the length of vectors?

a: Scaling matrix Explanation

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b: Orthogonal matrix Explanation

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c: Shear matrix Explanation

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d: Projection matrix Explanation

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28. If A is diagonalizable, then it must have...

a: Incorrect. Diagonalizable matrices may have complex eigenvalues. Explanation

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b: Correct. Diagonalizability requires a full set of independent eigenvectors. Explanation

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c: Incorrect. Determinant need not be zero. Explanation

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d: Incorrect. Columns need not be orthogonal. Explanation

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29. Which matrix is idempotent (A² = A)?

          1  0
          0  1

Explanation

          0  1
          0  0

Explanation

          1  0
          0  0

Explanation

          2  0
          0  2

Explanation

30. What does the nullity of a matrix refer to?

a: Number of rows Explanation

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b: Number of pivot columns Explanation

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c: Dimension of the null space Explanation

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d: Trace of the matrix Explanation

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31. If matrix A is singular, what does it imply?

a: det(A) ≠ 0 Explanation

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b: A has an inverse Explanation

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c: det(A) = 0 Explanation

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d: A has full rank Explanation

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32. The product of a matrix and its transpose is always...

a: Skew-symmetric Explanation

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b: Diagonal Explanation

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c: Symmetric Explanation

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d: Singular Explanation

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Understand various matrix operations, matrix decompositions, factorization and related operations