Mathematics

Linear Algebra

510 Questions

Linear algebra involves the study of matrices, vectors, and linear transformations. Common topics include finding the rank of a matrix, calculating eigenvalues and eigenvectors, and performing LU decomposition. These advanced mathematical concepts are frequently tested in engineering, statistics, and civil service examinations.

Matrix rank calculationLU decompositionEigenvalues and eigenvectorsMatrix invertibilityDeterminant properties

Linear Algebra Questions

Multiple choice

Given an orthogonal matrix $\begin{bmatrix} \ 1 & 1 & 1 & 1 \ \ 1 & 1 & -1 & -1 \ \ 1 & -1 & 0 & 0 \ \ 0 & 0 & 1 & -1 \end{bmatrix}$$\begin{bmatrix} \ AA^T \ \end{bmatrix}^{-1}$ is
  1. $\begin{bmatrix} \ \frac{1}{4} & 0 & 0 & 0 \\ \ 0 & \frac{1}{4} & 0 & 0 \\ \ 0 & 0 & \frac{1}{2} & 0 \\ \ 0 & 0 & 0 & \frac{1}{2} \\ \end{bmatrix}$
  2. $\begin{bmatrix} \ \frac{1}{2} & 0 & 0 & 0 \\ \ 0 & \frac{1}{2} & 0 & 0 \\ \ 0 & 0 & \frac{1}{2} & 0 \\ \ 0 & 0 & 0 & \frac{1}{2} \\ \end{bmatrix}$
  3. $\begin{bmatrix} \ 1 & 0 & 0 & 0 \\ \ 0 & 1 & 0 & 0 \\ \ 0 & 0 & 1 & 0 \\ \ 0 & 0 & 0 & 1 \\ \end{bmatrix}$
  4. $\begin{bmatrix} \ \frac{1}{4} & 0 & 0 & 0 \\ \ 0 & \frac{1}{4} & 0 & 0 \\ \ 0 & 0 & \frac{1}{4} & 0 \\ \ 0 & 0 & 0 & \frac{1}{4} \\ \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\text{From orthogonal matrix} \\ \begin{bmatrix} \ AA^T \ \end{bmatrix} = I \\ \text{Since the inverse of I is I, thus} \\ \begin{bmatrix} \ AA^T \ \end{bmatrix} = I^{-1} = I $

Multiple choice

The eigen values and the corresponding eigen vectors of a 2 × 2 matrix are given by

$\lambda_1$ = 8| V1 = $\begin{bmatrix} \ 1\ \ 1 \ \end{bmatrix}$| |$\lambda_2$ = 4| V1 = $\begin{bmatrix} \ 1\ \ -1 \ \end{bmatrix}$|
The matrix is
  1. $\begin{bmatrix} \ 6 & 2 \\ \ 2 & 6 \\ \end{bmatrix}$
  2. $\begin{bmatrix} \ 4 & 6 \\ \ 6 & 4 \\ \end{bmatrix}$
  3. $\begin{bmatrix} \ 2 & 4 \\ \ 4 & 2\\ \end{bmatrix}$
  4. $\begin{bmatrix} \ 4 & 8 \\ \ 8 & 4\\ \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Sum of the Eigen values must be equal to the sum of element of principal diagonal of matrix. Only matrix $\begin{bmatrix} \ 6 & 2 \ \ 2 & 6 \ \end{bmatrix}$ saisfy this condition.

Multiple choice

The rank of the matrix$\begin{bmatrix} \ 1& 1& 1 \ \ 1 & -1 & 0 \ \ 1 & 1 & 1 \ \end{bmatrix}$is

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\text{We have} \\ A = \begin{bmatrix} \ 1 & 1 & 1 \ \ 1 & -1 & 0\ 1 & 1 &1 \ \end{bmatrix} \sim \begin{bmatrix} \ 1 & 1 & 1 \ \ 1 & -1 & 0\ 0 & 0 &0 \ \end{bmatrix} \hspace{2cm} R_3 - R_1 \\ \text{Since one full row is zero, $\rho(A) < 3$} \\ \text{Now} \begin{vmatrix} \ 1 & 1 \ \ 1 & -1 \ 0 & 0 \ \end{vmatrix} = 2 \neq 0, thus \hspace{0.2cm}\rho(A) = 2$

Multiple choice

All the four entries of the 2 x 2 matrix = $\begin{bmatrix} \ p11 & p12 \ \ p21 & p22 \ \end{bmatrix}$ matrix are non-zero, and one of its eigen value is zero. Which of the following statements is true?

  1. p11p12 - p12p21 = 1

  2. p11p22 - p12p21 = -1

  3. p11p22 - p12p21 = 0

  4. p11p22 + p12p21 = 0
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The product of Eigen value is equal to the determinant of the matrix. Since one of the Eigen value is zero, the product of Eigen value is zero, thus determinant of the matrix is zero. Thus p11 p22 - p12 p21 = 0

Multiple choice

Consider the matrix P = $\begin{bmatrix} \ 0 & 1 \ \ -2 & -3 \ \end{bmatrix}$. The value of ep is
  1. $\begin{bmatrix} \ 2e^{-2}-3e^{-1} & e^{-1}-e^{-2} \\ \ 2e^{-2}-2e^{-1} & 5e^{-2}-e^-{1} \\ \end{bmatrix}$
  2. $\begin{bmatrix} \ e^{-1}+e^{-1} & 2e^{-2}-e^{-1} \\ \ 2e^{-1}-4e^{2} & 3e^{-1}+2e^{-2} \\ \end{bmatrix}$
  3. $\begin{bmatrix} \ 5e^{-2}-e^{-1} & 3e^{-1}-e^{-2} \\ \ 2e^{-2}-6e^{-1} & 4e^{-2}-6^{-1} \\ \end{bmatrix}$
  4. $\begin{bmatrix} \ 2e^{-1}-e^{-2} & e^{-1}-e^{-2} \\ \ -2e^{-1}+2e^{-2} & -e^{-1}+2e^{-2} \\ \end{bmatrix}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Multiple choice

The eigen values of a skew-symmetric matrix are

  1. always zero

  2. always purely imaginary

  3. either zero or purely imaginary

  4. always real
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Eigen values of a skew-symmetric matrix are either zero or pure imaginary occuring in conjugate pairs.

Multiple choice

Given A= $ \left[ \begin{array} \ 1 & 0 \\ 0 & 1 \end{array} \right] $ the state transition matrix eAt is given by
  1. $ \left[ \begin{array} \ 0 & e^{-t} \\\\ e^{-t} & 0 \end{array} \right] $
  2. $ \left[ \begin{array} \ e^{-t} & 0 \\\\ 0 & e^t & \end{array} \right] $
  3. $ \left[ \begin{array} \ e^{-t} & 0 \\\\ 0 & e^{-t} & \end{array} \right] $
  4. $ \left[ \begin{array} \ 0 & e^{t} \\\\ e^{t} & 0 \end{array} \right] $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Multiple choice

What are sparse matrices?

  1. Matrices with a relatively high proportion of zero entries

  2. Matrices with a relatively high proportion of non-zero entries

  3. Matrices with a relatively high proportion of 1's

  4. Matrices with a relatively high proportion of negative entries

  5. Matrices with all zero entries
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Matrices with a relatively high proportion of zero entries are called sparse matrices.