Vector Spaces and Matrix Operations - Matrix Theory: - Statistics Previous Year Questions

1

2019 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2019

The dimension of the vector space of \(7 \times 7\) real symmetric matrices with trace zero and the sum of the off-diagonal elements zero is

A

47

B

28

C

27

D

26

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2

2020 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2020

Let \(\mathbb{C}\) denote the set of all complex numbers. Consider the vector space \(V = \{(a,b,c) : a,b,c \in \mathbb{C}, \quad a + \bar{b} = 0, \quad b + \bar{c} = 0\}\), over the field of real numbers, where for any complex number \(z, \bar{z}\) denotes its complex conjugate. If \(i = \sqrt{-1}\), then a basis of \(V\) is

A

\(\{(1, -1, 1), (i, i, i)\}\)

B

\(\{(1, -1, 1), (i, -i, i)\}\)

C

\(\{(1, -i, 1), (i, 1, i)\}\)

D

\(\{(1, -i, 1), (i, 1, -1)\}\)

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3

2021 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2021

Let $A = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$ and $I_3$ be the $3 \times 3$ identity matrix. Then the nullity of $5A(I_3 + A + A^2)$ equals ________

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4

2021 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2021

Let \(A = [a \ u_1 \ u_2 \ u_3]\), \(B = [b \ u_1 \ u_2 \ u_3]\) and \(C = [u_2 \ u_3 \ u_1 \ a + b]\) be three \(4 \times 4\) real matrices, where \(a, b, u_1, u_2\) and \(u_3\) are \(4 \times 1\) real column vectors. Let \(\det(A)\), \(\det(B)\) and \(\det(C)\) denote the determinants of the matrices \(A\), \(B\) and \(C\), respectively. If \(\det(A) = 6\) and \(\det(B) = 2\), then \(\det(A + B) - \det(C)\) equals __________

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5

2022 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2022

Let \(M\) be any square matrix of arbitrary order \(n\) such that \(M^2 = 0\) and the nullity of \(M\) is 6. Then the maximum possible value of \(n\) (in integer) is ______

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6

2022 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2022

Consider the usual inner product in \(\mathbb{R}^4\). Let \(u \in \mathbb{R}^4\) be a unit vector orthogonal to the subspace
\(S = \{(x_1, x_2, x_3, x_4)^T \in \mathbb{R}^4 | x_1 + x_2 + x_3 + x_4 = 0\}\).
If \(v = (1, -2, 1, 1)^T\), and the vectors \(u\) and \(v - \alpha u\), \(\alpha \in \mathbb{R}\), are orthogonal, then the value of \(\alpha^2\) (rounded off to two decimal places) is equal to ______

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7

2023 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2023

Consider the following statements.
(I) Let \( A \) and \( B \) be two \( n \times n \) real matrices. If \( B \) is invertible, then \( rank(BA) = rank(A) \).
(II) Let \( A \) be an \( n \times n \) real matrix. If \( A^2x = b \) has a solution for every \( b \in \mathbb{R}^n \), then \( Ax = b \) also has a solution for every \( b \in \mathbb{R}^n \).
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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8

2023 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2023

Let \( A \) be a \( 2 \times 2 \) real matrix such that \( AB = BA \) for all \( 2 \times 2 \) real matrices \( B \). If trace of \( A \) equals 5, then determinant of \( A \) (rounded off to two decimal places) equals ______________

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9

2023 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2023

For any subset \(U\) of \(\mathbb{R}^n\), let \(L(U)\) denote the span of \(U\). For any two subsets \(T\) and \(S\) of \(\mathbb{R}^n\), which one of the following statements is NOT true?

A

If \(T\) is a proper subset of \(S\), then \(L(T)\) is a proper subset of \(L(S)\)

B

\(L(L(S)) = L(S)\)

C

\(L(T \cup S) = \{u + v : u \in L(T), v \in L(S)\}\)

D

If \(\alpha\), \(\beta\) and \(\gamma\) are three vectors in \(\mathbb{R}^n\) such that \(\alpha + 2\beta + 3\gamma = 0\), then \(L(\{\alpha, \beta\}) = L(\{\beta, \gamma\})\)

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10

2023 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2023

Let \( A \) be a \( 3 \times 3 \) real matrix such that \( A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ 0 \end{bmatrix} \), \( A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ 0 \end{bmatrix} \) and \( A \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 4 \end{bmatrix} \). Then which of the following statements is/are true?

A

\(A \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 2 \\ -2 \end{bmatrix} \)

B

\(A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ -2 \\ 2 \end{bmatrix} \)

C

\(A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \\ 2 \end{bmatrix} \)

D

\(A \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 8 \\ 4 \\ 0 \end{bmatrix} \)

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11

2023 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2023

Consider the orthonormal set \( \left\{ v_1 = \begin{bmatrix} \frac{1}{\sqrt{3}} \\ -\frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}} \end{bmatrix}, v_2 = \begin{bmatrix} \frac{1}{\sqrt{6}} \\ \frac{2}{\sqrt{6}} \\ \frac{1}{\sqrt{6}} \end{bmatrix}, v_3 = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ -\frac{1}{\sqrt{2}} \end{bmatrix} \right\} \) with respect to the standard inner product on \( \mathbb{R}^3 \). If \( u = \begin{bmatrix} a \\ b \\ c \end{bmatrix} \) is the vector such that inner products of \( u \) with \( v_1, v_2 \) and \( v_3 \) are \( 1, 2 \) and \( 3 \), respectively, then \( a^2 + b^2 + c^2 \) (in integer) equals __________

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12

2024 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2024

Let \(\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3, \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) be vectors in \(\mathbb{R}^4\). Let \(U\) be the span of \(\{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\}\) and let \(V\) be the span of \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\).
Consider the following statements:
(I) If the dimension of \(U \cap V\) is 2 and the dimension of \(U\) is 3, then \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is linearly dependent.
(II) If \(U + V = \{\mathbf{u} + \mathbf{v} : \mathbf{u} \in U, \mathbf{v} \in V\} = \mathbb{R}^4\), then either \(\{\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3\}\) is linearly independent or \(\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}\) is linearly independent.
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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13

2024 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2024

Consider \(\mathbb{R}^2\) with standard inner product. If \(\mathbf{u} = \begin{bmatrix} a \\ b \end{bmatrix}\) is the vector in \(\mathbb{R}^2\) such that the inner product of \(\mathbf{u}\) with \(\begin{bmatrix} 1 \\ 2 \end{bmatrix}\) is 2 and with \(\begin{bmatrix} 4 \\ -2 \end{bmatrix}\) is \(-1\), then which one of the following statements is true?

A

Inner product of \(\mathbf{u}\) with \(\begin{bmatrix} 1 \\ -1 \end{bmatrix}\) is \(\frac{1}{2}\)

B

Inner product \(\mathbf{u}\) with \(\begin{bmatrix} -1 \\ 1 \end{bmatrix}\) is \(\frac{3}{5}\)

C

Inner product of \(\mathbf{u}\) with \(\begin{bmatrix} 2 \\ -2 \end{bmatrix}\) is \(-\frac{6}{5}\)

D

Inner product of \(\mathbf{u}\) with \(\begin{bmatrix} -2 \\ 1 \end{bmatrix}\) is \(\frac{4}{5}\)

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14

2024 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2024

Let \( A = \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{bmatrix} \) be a \( 2 \times 3 \) real matrix, where \( (a_1, a_2, a_3) \neq (0,0,0) \) and \( (b_1, b_2, b_3) \neq (0,0,0) \). Assume that the rank of \( A \) is 1. Define the subspaces \( W = \left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : A\mathbf{x} = \mathbf{0} \right\} \), \( W_1 = \left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : a_1 x_1 + a_2 x_2 + a_3 x_3 = 0 \right\} \), and \( W_2 = \left\{ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \in \mathbb{R}^3 : b_1 x_1 + b_2 x_2 + b_3 x_3 = 0 \right\} \). Consider the following statements:
(I) \( W = W_1 \cap W_2 \)
(II) \( W_1 = W_2 \)
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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15

2025 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2025

Let \(S = \{(x, y, z) \in \mathbb{R}^3 \setminus \{(0,0,0)\} : z = -(x + y)\}\). Denote \(S^\perp = \{(p, q, r) \in \mathbb{R}^3 : px + qy + rz = 0 \text{ for all } (x, y, z) \in S\}\). Then which one of the following options is correct?

A

\(S^\perp\) is not a subspace of \(\mathbb{R}^3\)

B

\(S^\perp = \{(0,0,0)\}\)

C

\(\dim(S^\perp) = 1\)

D

\(\dim(S^\perp) = 2\)

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16

2025 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2025

Let \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear map defined by \[ T(x_1, x_2, x_3) = (3x_1 + 5x_2 + x_3, \, x_3, \, 2x_1 + 2x_3). \] Then the rank of \( T \) is equal to ______________ (answer in integer).

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17

2026 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2026

Consider the following two subspaces of \( \mathbb{R}^4 \):
\( W_1 = \{(x_1, x_2, x_3, x_4) : x_1 + x_2 + x_3 + x_4 = 0 \} \)
\( W_2 = \{(x_1, x_2, x_3, x_4) : x_1 + 2x_2 + 3x_3 + 4x_4 = 0 \} \).
Which of the following statements is correct?

A

\(\dim(W_1 \cap W_2) = 0 \)

B

\(\dim(W_1 \cap W_2) = 1 \)

C

\(\dim(W_1 \cap W_2) = 2 \)

D

\(\dim(W_1 \cap W_2) = 3 \)

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18

2026 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2026

Consider the system of linear equations
\[x + y + z = 1\]
\[2x + y + 3z = 6\]
\[3x + 2y + kz = k + 1\]
If the above system has no solution, then the value of \(k\) equals ______________ (answer in integer).

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19

2026 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2026

Consider the subspace \[U = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 + x_2 + x_4 = 0, x_3 = 0\}.\] Let \(U^\perp\) be its orthogonal complement. The value of \(\dim(U^\perp)\) equals ______________ (answer in integer).

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20

2026 · Statistics · Matrix Theory: · Vector Spaces and Matrix Operations

Statistics (ST) 2026

Consider the system of equations
\[ x + 2y - z = a \]
\[ x + y + 3z = b \]
\[ 2x + 3y + 2z = c \]
Consider the following statements:
(I) For every \( (a,b,c) \in \mathbb{R}^3 \), the above system has a solution.
(II) For \( (a,b,c) = (0,0,0) \), the solution set is given by \( \{(-7t, \; 4t, \; t) : t \in \mathbb{R}\} \).
Which of the following statements is correct?

A

Only statement (I) is correct

B

Only statement (II) is correct

C

Both statements (I) and (II) are correct

D

Neither statement (I) nor statement (II) is correct

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