Linear Algebra · Engineering Mathematics · GATE EE

Two $n \times n$ matrices $A$ and $B$ have a common eigenvalue 2 , and the same corresponding nonzero eigenvector. Which of the following options is/are correct?

(Note: $I$ is the $n \times n$ identity matrix.)

A is an $m \times m$ skew-symmetric matrix with real-valued entries, and $x$ is an $m$-dimensional column vector with real-valued entries such that $x^T x=1$. The quantity $x^T A x$ evaluates to $\_\_\_\_$ . (Answer in integer)

The relationship between two variables $x$ and $y$ is given by $x+p y+q=0$ and is shown in the figure. Find the values of $p$ and $q$.

Note: The figure shown is representative.

Let $A=\left[\begin{array}{ccc}1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1\end{array}\right]$ and $b=\left[\begin{array}{c}1 / 3 \\ -1 / 3 \\ 0\end{array}\right]$, then the system of linear equations $A x=b$ has

Let $P=\left[\begin{array}{ccc}2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$ and let $I$ be the identity matrix. Then $P^2$ is equal to

Which one of the following matrices has an inverse?

The sum of the eigenvalues of the matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2$ is ______ (rounded off to the nearest integer).

For a given vector $${[\matrix{ 1 & 2 & 3 \cr } ]^T}$$, the vector normal to the plane defined by $${w^T}x = 1$$ is

In the figure, the vectors u and v are related as : Au = v by a transformation matrix A. The correct choice of A is

Consider a 3 $$\times$$ 3 matrix A whose (i, j)-th element, a_i,j = (i $$-$$ j)³. Then the matrix A will be

Let $p$ and $q$ be real numbers such that $p^2+q^2=1$. The eigen values of the matrix $\left[\begin{array}{cc}p & q \\ q & -p\end{array}\right]$ are

$$P:$$ There is a unique solution.
$$Q:$$ The equations are linearly independent .
$$R:$$ All eigen values of the coefficient matrix are non zero .
$$S:$$ The determinant of the coefficient matrix is non-zero .

Which one of the following statements is ALWAYS correct about a collection of $p$ column vectors, each having $n$ real-valued entries?

Consider the system of linear equations: $A x=b$, where $A$ is an $\mathrm{n} \times \mathrm{n}$ matrix, and $x$ and $b$ are $n$-dimensional column vectors.

Suppose this system of equations has a unique solution. Which of the following statements is/are correct?

e⁴ denotes the exponential of a square matrix A. Suppose $$\lambda$$ is an eigen value and v is the corresponding eigen-vector of matrix A.

Consider the following two statements:

Statement 1 : e^$$\lambda$$ is an eigen value of e^A.

Statement 2 : v is an eigen-vector of e^A.

Which one of the following options is correct?

Consider a matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 2} \cr 0 & 1 & 1 \cr } } \right]$$. The matrix A satisfies the equation 6A^$$-$$1 = A² + cA + dI, where c and d are scalars and I is the identify matrix. Then (c + d) is equal to

Let $A$ be a $10 \times 10$ matrix such that $A^5$ is null matrix and let $I$ be the $10 \times 10$ identity matrix. The determinant of $A+I$ is $\_\_\_\_$ .

The absolute value of the ratio of the maximum eigenvalue to the minimum eigenvalue is ___________.