Linear Algebra · Engineering Mathematics · GATE CE

For the matrix $[\mathrm{A}]$ given below, the transpose is $\qquad$ .

$$ [A]=\left[\begin{array}{lll} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{array}\right] $$

Suppose $\lambda$ is an eigenvalue of matrix A and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $\mathrm{B}=\mathrm{A}-2 \mathrm{I}$, where I is the identity matrix. Then, the eigenvalue of B corresponding to the eigenvector $x$ is equal to

Let $A=\left[\begin{array}{cc}1 & 1 \\ 1 & 3 \\ -2 & -3\end{array}\right]$ and $b=\left[\begin{array}{l}b_1 \\ b_2 \\ b_3\end{array}\right]$. For $\mathrm{Ax}=\mathrm{b}$ to be solvable, which one of the following options is the correct condition on $b_1, b_2$ and $b_3$ :

The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication.

P: $(A + B)^T = A^T + B^T$

Q: $(AB)^T = B^T A^T$

Which one of the following options is CORRECT?

For the matrix

$\rm [A]=\begin{bmatrix}1&-1&0\\\ -1&2&-1\\\ 0&-1&1\end{bmatrix}$

which of the following statements is/are TRUE?

Let y be a non-zero vector of size 2022 $$\times$$ 1. Which of the following statements is/are TRUE?

The components of pure shear strain in a sheared are given in the matrix form:

$$\varepsilon = \left[ {\matrix{ 1 & 1 \cr 1 & { - 1} \cr } } \right]$$

Here, Trace ($$\varepsilon $$) = 0. Given, P = Trace ($$\varepsilon$$⁸) and Q = Trace ($$\varepsilon $$¹¹).

The numerical value of (P + Q) is ___________. (in integer)

P and Q are two square matrices of the same order. Which of the following statements is/are correct?

The matrix M is defined as

$$M = \left[ {\matrix{ 1 & 3 \cr 4 & 2 \cr } } \right]$$

and has eigenvalues 5 and $$-$$2. The matrix Q is formed as

Q = M³ $$-$$ 4M² $$-$$ 2M

The Cartesian coordinates of a point P in a right-handed coordinate system are (1, 1, 1). The transformed coordinates of P due to a 45$$^\circ$$ clockwise rotation of the coordinate system about the positive x-axis are

The characteristic equation for these simultaneous equation is

This system of equations has

$$(I)$$ The maximum number of linearly independent column vectors of a matrix $$A$$ is called the rank of $$A.$$

$$(II)$$ If $$A$$ is $$nxn$$ square matrix then it will be non-singular if rank of $$A=n$$

Pick the CORRECT eigen value(s) of the matrix $[\mathrm{A}]$ from the following choices.

$$ [A]=\left[\begin{array}{ll} 6 & 8 \\ 4 & 2 \end{array}\right] $$

Consider two matrices $A = \begin{bmatrix}2 & 1 & 4 \\ 1 & 0 & 3\end{bmatrix}$ and $B = \begin{bmatrix}-1 & 0 \\ 2 & 3 \\ 1 & 4 \end{bmatrix}$.

The determinant of the matrix $AB$ is __________ (in integer).

What are the eigenvalues of the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ ?

Cholesky decomposition is carried out on the following square matrix [𝐴]. 

$\rm [A]=\begin{bmatrix}8&-5\\\ -5&a_{22}\end{bmatrix}$

Let 𝑙_ij and 𝑎ij be the (i, j)^th elements of matrices [𝐿] and [𝐴], respectively. If the element 𝑙₂₂ of the decomposed lower triangular matrix [𝐿] is 1.968, what is the value (rounded off to the nearest integer) of the element 𝑎₂₂?

For the matrix

$[A]= \begin{bmatrix}1&2&3\\\ 3&2&1\\\ 3&1&2 \end{bmatrix} $

which of the following statements is/are TRUE?