Eigenvalues, Diagonalization and Quadratic Forms - Matrix Theory: - Statistics Previous Year Questions

1

2019 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2019

Let \(A\) be a \(6 \times 6\) complex matrix with \(A^3 \neq \mathbf{0}\) and \(A^4 = \mathbf{0}\). Then the number of Jordan blocks of \(A\) is

A

1 or 6

B

2 or 3

C

4

D

5

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2

2019 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2019

The minimal polynomial of the matrix \(\begin{bmatrix} 1 & 1 & 2 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{bmatrix}\) is

A

\((x - 1)(x - 2)\)

B

\((x - 1)^2(x - 2)\)

C

\((x - 1)(x - 2)^2\)

D

\((x - 1)^2(x - 2)^2\)

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3

2019 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2019

The matrix \(\begin{bmatrix} 1 & x & z \\ 0 & 2 & y \\ 0 & 0 & 1 \end{bmatrix}\) is diagonalizable when \((x, y, z)\) equals

A

\((0,0,1)\)

B

\((1,1,0)\)

C

\((\sqrt{2}, \sqrt{2}, 2)\)

D

\((\sqrt{2}, \sqrt{2}, \sqrt{2})\)

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4

2019 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2019

Let \( A \) be a \( n \times n \) positive semi-definite matrix with eigenvalues \( \lambda_1 \geq \cdots \geq \lambda_n \), and with \( \alpha \) as the maximum diagonal entry. We can find a vector \( x \) such that \( x^T x = 1 \), where \( t \) denotes the transpose, and

A

\(x^T A x > \lambda_1 \)

B

\(x^T A x < \lambda_n \)

C

\(\lambda_n \leq x^T A x \leq \lambda_1 \)

D

\(x^T A x > n \alpha \)

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5

2019 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2019

Let \(I\) be the \(4 \times 4\) identity matrix and \(v = (1, 2, 3, 4)^t\), where \(t\) denotes the transpose. Then the determinant of \(I + vv^t\) is equal to ...

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6

2020 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2020

Let \(M\) be a \(3 \times 3\) non-zero idempotent matrix and let \(I_3\) denote the \(3 \times 3\) identity matrix. Then which of the following statements is FALSE?

A

The eigenvalues of \(M\) are 0 and 1

B

\(\text{Rank}(M) = \text{Trace}(M)\)

C

\(I_3 - M\) is idempotent

D

\((I_3 + M)^{-1} = I_3 - 2M\)

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7

2020 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2020

Consider the linear transformation \( T: \mathbb{C}^3 \to \mathbb{C}^3 \) defined by
\[ T((x, y, z)) = \left( x, \; \frac{\sqrt{3}}{2} y - \frac{1}{2} z, \; \frac{1}{2} y + \frac{\sqrt{3}}{2} z \right), \]
where \( \mathbb{C} \) is the set of all complex numbers and \( \mathbb{C}^3 = \mathbb{C} \times \mathbb{C} \times \mathbb{C} \). Which of the following statements is TRUE?

A

There exists a non-zero vector \(X \) such that \(T(X) = -X \)

B

There exist a non-zero vector \(Y \) and a real number \(\lambda \neq 1 \) such that \(T(Y) = \lambda Y \)

C

\(T \) is diagonalizable

D

\(T^2 = I_3 \), where \(I_3 \) is the \(3 \times 3 \) identity matrix

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8

2020 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2020

Let \( M \) be a \( 4 \times 4 \) matrix with \( (x - 1)^2 (x - 3)^2 \) as its minimal polynomial. Then, which of the following statements is FALSE?

A

The eigenvalues of \(M \) are 1 and 3

B

The algebraic multiplicity of the eigenvalue 1 is 3

C

\(M \) is NOT diagonalizable

D

\(\text{Trace}(M) = 8 \)

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9

2021 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2021

Let $A$ be the $2 \times 2$ real matrix having eigenvalues $1$ and $-1$, with corresponding eigenvectors $\begin{bmatrix} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{bmatrix}$ and $\begin{bmatrix} -\frac{1}{2} \\ \frac{\sqrt{3}}{2} \end{bmatrix}$, respectively. If $A^{2021} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $a + b + c + d$ equals ________ (round off to 2 decimal places).

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10

2021 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2021

Let \( M \) be the collection of all \( 3 \times 3 \) real symmetric positive definite matrices. Consider the set \[ S = \left\{ A \in M : A^{50} - \frac{1}{4} A^{48} = 0 \right\}, \] where \( 0 \) denotes the \( 3 \times 3 \) zero matrix. Then the number of elements in \( S \) equals

A

0

B

1

C

8

D

\(\infty \)

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11

2021 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2021

Let \( A \) be a \( 3 \times 3 \) real matrix such that \( I_3 + A \) is invertible and let \( B = (I_3 + A)^{-1} (I_3 - A) \), where \( I_3 \) denotes the \( 3 \times 3 \) identity matrix. Then which one of the following statements is true?

A

If \(B \) is orthogonal, then \(A \) is invertible

B

If \(B \) is orthogonal, then all the eigenvalues of \(A \) are real

C

If \(B \) is skew-symmetric, then \(A \) is orthogonal

D

If \(B \) is skew-symmetric, then the determinant of \(A \) equals -1

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12

2021 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2021

Let \( Y \) follow \( N_8(\underline{0}, I_8) \) distribution, where \( I_8 \) is the \( 8 \times 8 \) identity matrix. Let \( \underline{Y}^T \Sigma_1 \underline{Y} \) and \( \underline{Y}^T \Sigma_2 \underline{Y} \) be independent and follow central chi-square distributions with \( 3 \) and \( 4 \) degrees of freedom, respectively, where \( \Sigma_1 \) and \( \Sigma_2 \) are \( 8 \times 8 \) matrices and \( \underline{Y}^T \) denotes transpose of \( \underline{Y} \). Then which of the following statements is/are true?
P : \( \Sigma_1 \) and \( \Sigma_2 \) are idempotent.
Q : \( \Sigma_1 \Sigma_2 = \underline{0} \), where \( \underline{0} \) is the \( 8 \times 8 \) zero matrix.

A

P only

B

Q only

C

Both P and Q

D

Neither P nor Q

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13

2022 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2022

Let \( M \) be a \( 2 \times 2 \) real matrix such that \( (I + M)^{-1} = I - \alpha M \), where \( \alpha \) is a non-zero real number and \( I \) is the \( 2 \times 2 \) identity matrix. If the trace of the matrix \( M \) is 3, then the value of \( \alpha \) is

A

\(\frac{3}{4} \)

B

\(\frac{1}{3} \)

C

\(\frac{1}{2} \)

D

\(\frac{1}{4} \)

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14

2022 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2022

Let \( M \) be a \( 3 \times 3 \) real symmetric matrix with eigenvalues \( -1, 1, 2 \) and the corresponding unit eigenvectors \( u, v, w \), respectively. Let \( x \) and \( y \) be two vectors in \( \mathbb{R}^3 \) such that \( Mx = u + 2(v + w) \) and \( M^2y = u - (v + 2w) \). Considering the usual inner product in \( \mathbb{R}^3 \), the value of \( |x + y|^2 \), where \( |x + y| \) is the length of the vector \( x + y \), is

A

1.25

B

0.25

C

0.75

D

1

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15

2022 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2022

Let \(M\) be any \(3 \times 3\) symmetric matrix with eigenvalues 1, 2 and 3. Let \(N\) be any \(3 \times 3\) matrix with real eigenvalues such that \(MN + N^T M = 3I\), where \(I\) is the \(3 \times 3\) identity matrix. Then which of the following cannot be eigenvalue(s) of the matrix \(N\)?

A

\(\frac{1}{4}\)

B

\(\frac{3}{4}\)

C

\(\frac{1}{2}\)

D

\(\frac{7}{4}\)

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16

2022 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2022

Let M be a 3 × 2 real matrix having a singular value decomposition as M = USVT, where the matrix S = , U is a 3 × 3 orthogonal matrix, and V is a 2 × 2 orthogonal matrix. Then which of the following statements is/are true?

A

The rank of the matrix M is 1

B

The trace of the matrix MᵀM is 4

C

The largest singular value of the matrix (MᵀM)⁻¹Mᵀ is 1

D

The nullity of the matrix M is 1

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17

2023 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2023

Let \( A \) be a \( 3 \times 3 \) real matrix having eigenvalues \( 1, 0 \), and \( -1 \). If \( B = A^2 + 2A + I_3 \), where \( I_3 \) is the \( 3 \times 3 \) identity matrix, then which one of the following statements is true?

A

\(B^3 - 5B^2 + 4B = 0 \)

B

\(B^3 - 5B^2 - 4B = 0 \)

C

\(B^3 + 5B^2 - 4B = 0 \)

D

\(B^3 + 5B^2 + 4B = 0 \)

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18

2023 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2023

Let \( A \) be an \( n \times n \) real matrix. Consider the following statements.
(I)    If \( A \) is symmetric, then there exists \( c \geq 0 \) such that \( A + cI_n \) is symmetric and positive definite, where \( I_n \) is the \( n \times n \) identity matrix.
(II)    If \( A \) is symmetric and positive definite, then there exists a symmetric and positive definite matrix \( B \) such that \( A = B^2 \).
Which of the above statements is/are true?

A

Only (I)

B

Only (II)

C

Both (I) and (II)

D

Neither (I) nor (II)

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19

2024 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2024

Let \( A \) be a \( 3 \times 3 \) real matrix and let \( I_3 \) be the \( 3 \times 3 \) identity matrix. Which one of the following statements is NOT true?

A

If the row-reduced echelon form of \(A \) is \(I_3 \), then zero is not an eigenvalue of \(A \)

B

If zero is not an eigenvalue of \(A \), then the row-reduced echelon form of \(A \) is \(I_3 \)

C

If \(A \) has three distinct eigenvalues, then the row-reduced echelon form of \(A \) is \(I_3 \)

D

If the system of equations \(A\mathbf{x} = \mathbf{b} \) has a solution for every \(3 \times 1 \) real column vector \(\mathbf{b} \), then the row-reduced echelon form of \(A \) is \(I_3 \)

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20

2024 · Statistics · Matrix Theory: · Eigenvalues, Diagonalization and Quadratic Forms

Statistics (ST) 2024

Let \( A \) be an \( n \times n \) real matrix. Which of the following statements is/are true?

A

If \(A \) is a symmetric matrix such that \(A + \epsilon I_n \) is positive semi-definite for every \(\epsilon > 0 \), where \(I_n \) is the \(n \times n \) identity matrix, then \(A \) is positive semi-definite

B

If \(n \) is odd, then \(A - A^T \) is not invertible

C

If \(A \) is a symmetric matrix such that the singular values of \(A \) are all strictly positive, then \(A \) is positive definite

D

If 1 is the only singular value of \(A \), then \(A \) is orthogonal

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