Linear Algebra

Subtopic: Matrices & Determinants

The number of independent elastic constants that a fully anisotropic linear elastic material can have is ______.

Subtopic: Matrices & Determinants

An n × n square matrix A satisfies AT = A-1. The determinant of this matrix may take which of the following value(s)?

Subtopic: Matrices & Determinants

The state of stress at a point in a^2-D body, in the x-y Cartesian coordinate system, is represented in matrix form as [$\sigma$]. The transformation matrix [Q] rotates the coordinate system to a new x'-y' Cartesian coordinate system. Select the CORRECT option(s) that represent(s) the state of stress in the new coordinate system.

Subtopic: Matrices & Determinants

If a matrix can be written as A = uvT, where both u and v are n-dimensional real- valued non-zero column vectors, then the rank of the matrix A is ________ (answer in integer).

Subtopic: Eigenvalues & Eigenvectors

For the matrix A = $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, if the relation a + b = c + d holds and a, b, c, d ≠ 0, then which one of the following statements about A is FALSE?

Subtopic: Matrices & Determinants

For any real symmetric matrix A, the transpose of A is ____ .

Subtopic: Systems of Linear Equations

2x + 3y + z = 0 x + y = 0 y + z = 0 The given system of equations has

Subtopic: Eigenvalues & Eigenvectors

The eigenvalues of the matrix [1 2 0 3] are _____.

Subtopic: Systems of Linear Equations

The following system of linear equations 7x - 3y + z = 0 3x - y + z = 0 x - y - z = 0 has:

Subtopic: Systems of Linear Equations

Given y = epx sin qx, where p and q are non-zero real numbers, the value of the differential expression $\frac{d^2y}{dx^2}$ - 2p $\frac{dy}{dx}$ + (p^2 + q^2)y is

Subtopic: Eigenvalues & Eigenvectors

Consider the matrix A = [5 -4 k -1], where k is a constant. If the determinant of A is 3, then the ratio of the largest eigenvalue of A to the constant k is ___________ (rounded off to 1 decimal place).

Subtopic: Eigenvalues & Eigenvectors

For a general aviation airplane, one of the complex conjugate pair of eigenvalues for longitudinal dynamics is given by -0.039 ± 0.0567 i (in SI units) . If the system is disturb ed to excite only this mode, the time taken for the amplitude of response to become half in magnitude is _______ s (rounded off to 1 decimal place).

Subtopic: Eigenvalues & Eigenvectors

If a monotonic and continuous function ()y f x= has only one root in the interval 12x x x , then

Subtopic: Eigenvalues & Eigenvectors

Which of the following statement(s) is/are true with respect to eigenvalues and eigenvectors of a matrix?

Subtopic: Systems of Linear Equations

Given the function ( ) ( 3)( 2)y x x x= + - , for 44 x-   . What is the value of x at which the function has a minimum?

Subtopic: Eigenvalues & Eigenvectors

Let $\hat{i}, \hat{j}$, and $\hat{k}$ be the unit vectors in the $x, y$ and $z$ directions, respectively. If the vector $\hat{i} + \hat{j}$ is rotated about positive $\hat{k}$ by $135^\circ$, one gets:

Subtopic: Systems of Linear Equations

The height of a right circular cone of maximum volume that can be enclosed within a hollow sphere of radius R is

Subtopic: Eigenvalues & Eigenvectors

The largest eigenvalue of the given matrix is _____: $\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$

Subtopic: Eigenvalues & Eigenvectors

A scalar field $u(x, y)$ is governed by the equation $\frac{\partial^2 u}{\partial x^2} + 6\frac{\partial^2 u}{\partial x \partial y} + 9\frac{\partial^2 u}{\partial y^2} = x + 2y$. The mathematical nature of this equation is:

Subtopic: Systems of Linear Equations

In an aerospace engineering analysis of Linear Algebra (Systems of Linear Equations), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Subtopic: Eigenvalues & Eigenvectors

For the Gaussian normal distribution function $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-z}$, where $z = \frac{(x - \mu)^2}{2\sigma^2}$, and $\sigma, \mu$ are constants, the maximum of $f(x)$ occurs at:

Subtopic: Eigenvalues & Eigenvectors

Given $A = \begin{bmatrix} \sin\theta & \tan\theta \\ 0 & \cos\theta \end{bmatrix}$, the sum of squares of the eigenvalues of $A$ is:

Subtopic: Matrices & Determinants

A load P is applied to the free end of a stepped cantilever beam as shown in the figure. The Young's modulus of the material is E, and the moments of inertia of the two sections of length 2 m and 1 m are / and 3l, respectively. Ignoring transverse shear and stress concentration effects, the deflection at the point where the load is applied at the free end of the cantilever is 2m - Im 31 = 3N 23 243 EI 1 3EI 43 3EI 23 ЗЕІ The three dimensional strain-stress relation for an isotropic material, written in a general matrix form, is C C 0 Vy= Y xe Yxy) C C 0 A C 0 C A 0 0 0 0 0 0 0 0 0 0 0 0 0 Ty= 0 BJl [xy A, B and C are compliances which depend on the elastic properties of the material. Which one of the following is correct? c= 4 - B c=1+ B 2 2

Subtopic: Matrices & Determinants

In the equation AX = B, A = 1 0 0 11 X= B-?. , where A is -V=] (round off to one an orthogonal matrix, the sum of the unknowns, *+ y+z=_ decimal place). If f(x* - 2x+ 1) dr is evaluated numerically using trapezoidal rule with four intervals, the difference between the numerically evaluated value and the analytical value of the integral is equal to (round off to three decimal places). O.No. 37

Subtopic: Systems of Linear Equations

In an aerospace engineering analysis of Linear Algebra (Systems of Linear Equations), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?

Subtopic: Systems of Linear Equations

The maximum value of the function f(x) = xe-* (where x is real) is

Subtopic: Eigenvalues & Eigenvectors

Vector b is obtained by rotating a = î + j by 90º about k, where i, j and k are unit vectors along the x, y and z axes, respectively. b is given by

Subtopic: Eigenvalues & Eigenvectors

One of the eigenvalues of the matrix $M = \begin{bmatrix} 1 & 3 \\ 0 & 2 \end{bmatrix}$ is 1. The other eigenvalue is _________.

Subtopic: Matrices & Determinants

The determinant of the matrix [ 1 1 -1 2 1 0 3 1 1 ] is ______ (accurate to one decimal place).

Subtopic: Matrices & Determinants

The determinant of the matrix $M = \begin{bmatrix} 1 & 1 & -1 \\ 2 & 1 & 0 \\ 3 & 1 & 1 \end{bmatrix}$ is:

Subtopic: Systems of Linear Equations

Consider the vector field v⃗ = - y r2 î + x r2 ĵ ; where r = √x^2 + y^2. The contour integral ∮ v⃗ ⋅ ds⃗⃗⃗⃗⃗, where ds⃗⃗⃗⃗⃗ is tangent to the contour that encloses the origin, is __________ (accurate to two decimal places).

Subtopic: Systems of Linear Equations

Expansion with finite pressure difference with the surrounding. (i1) Compression with infinitesimal pressure difference with the surrounding. (iii) Heat transfer with finite temperature difference with the reservoir. (IV) Heat transfer with infinitesimal temperature difference with the reservoir. Out of these which processes are reversible?

Subtopic: Matrices & Determinants

(4) Matrix A = and vector (6) = (5) are given. If vector {x) is the solution to the system of equations [A]{x} = {b}, which of the following is true for {x}:

Subtopic: Matrices & Determinants

Let matrix [A] = [a^2] Then for any non-trivial vector (x) = (xz) which of the following is true for the value of K = {x} [A]{x}:

Subtopic: Systems of Linear Equations

Consider the following system of linear equations: 2x - y + z = 1 3x - 3y + 4z = 6 x - 2y + 3z = 4 This system of linear equations has

Subtopic: Eigenvalues & Eigenvectors

If all the eigenvalues of a matrix are real and equal, then

Subtopic: Systems of Linear Equations

An aerospace system operating under standard sea level conditions is governed by Linear Algebra (Systems of Linear Equations). Compute the required parameter in standard SI units (rounded off to the acceptable range).

Subtopic: Matrices & Determinants

Values of a, b and c, which render the matrix 11 32 1 0 3 11 32 a Qb c        orthonormal are, respectively

Subtopic: Systems of Linear Equations

The value of k for which the system of equations 21 ; 28 3xy k z x k yz+ += ++= has no solution is

Subtopic: Eigenvalues & Eigenvectors

One eigenvalue of the matrix 271 0 522 5 16 5 A ⎡ ⎤ ⎢ ⎥=⎢ ⎥ ⎢ ⎥⎣ ⎦ is - 9.33. One of the other eigenvalues is

Subtopic: Eigenvalues & Eigenvectors

) ( ) ( ) 1 ( cos 1 cossin k kk k k x x x xx   (D) ) ( ) ( ) ( ) 1 ( cos 1 cossin k kk k k x x x xx   Consider the matrix where a and b are real numbers. The two eigenvalues of this m atrix     2 2 b a 1 and 2 are real and distinct  2 1  when

Subtopic: Eigenvalues & Eigenvectors

An aircraft in level flight encounters a vertical gust, which excites the phugoid mode. The phugoid motion completes 10 cycles in 50 s and its amplitude reduces to half of its maximum value in 25 s. The eigenvalues of the phugoid mode are

Subtopic: Systems of Linear Equations

The radial and azimuthal velocities on the cylinder surface at 8 = I / 2 are

Subtopic: Matrices & Determinants

The linearized dynamics of an aircraft (which has no large rotating components) in straight and level flight is governed by the equations dx dt [[a] [в]]. [[c] [D] where x=[u w q 0 v pr ], [I represents the transpose of a matrix, [A], [B], [C] and [D] are 4x4 matrices and [O] is the 4x4 null matrix. Which of the following is true ?

Subtopic: Eigenvalues & Eigenvectors

The pair of eigenvalues that represent the phugoid mode is

Subtopic: Systems of Linear Equations

A normal to the surface is

Subtopic: Eigenvalues & Eigenvectors

The value of the line integral (x dy - y dx) taken anticlockwise along a circle of unit radius 2л is

Subtopic: Eigenvalues & Eigenvectors

If the vertical tail of an airplane is inverted and put below the horizontal tail, then the contribution to roll derivative, 2C, will be

Subtopic: Systems of Linear Equations

Let a system of linear equations be as follows: x-y+2z=0 2x + 3y -z = 0 2x -2y + 4z = 0 This system of equations has

Subtopic: Systems of Linear Equations

An athlete starts running with a speed ½. Subsequently, his speed decreases by an amount that is proportional to the distance that he has already covered. The distance covered will be