Linear Algebra · Engineering Mathematics · GATE ME

Consider the system of linear equationsx + 2y + z = 52x + ay + 4z = 122x + 4y + 6z = bThe values of a and b such that there exists a non-trivial null ...

A linear transformation maps a point (𝑥, 𝑦) in the plane to the point (𝑥̂, 𝑦̂) according to the rule 𝑥̂ = 3𝑦, 𝑦̂ = 2𝑥. Then, the disc 𝑥2 + ...

If A = $\begin{bmatrix} 10 & 2k + 5 \\\ 3k - 3 & k + 5 \end{bmatrix} $ is a symmetric matrix, the value of k is _______....

The determinant of a $$2 \times 2$$ matrix is $$50.$$ If one eigenvalue of the matrix is $$10,$$ the other eigenvalue is __________.

The product of eigenvalues of the matrix $$P$$ is $$P = \left[ {\matrix{ 2 & 0 & 1 \cr 4 & { - 3} & 3 \cr 0 & 2 &...

The condition for which the eigenvalues of the matrix $$A = \left[ {\matrix{ 2 & 1 \cr 1 & k \cr } } \right]$$ are positive, is

A real square matrix $$A$$ is called skew-symmetric if

The solution to the system of equations is $$\left[ {\matrix{ 2 & 5 \cr { - 4} & 3 \cr } } \right]\left\{ {\matrix{ x \cr ...

The lowest eigen value of the $$2 \times 2$$ matrix $$\left[ {\matrix{ 4 & 2 \cr 1 & 3 \cr } } \right]$$ is ______.

At least one eigenvalue of a singular matrix is

If any two columns of a determinant $$P = \left| {\matrix{ 4 & 7 & 8 \cr 3 & 1 & 5 \cr 9 & 6 & 2 \cr } } \rig...

Which one of the following equations is a correct identity for arbitrary $$3 \times 3$$ real matrices $$P,Q$$ and $$R$$?

Consider a $$3 \times 3$$ real symmetric matrix $$S$$ such that two of its eigen values are $$a \ne 0,$$ $$b\,\, \ne 0$$ with respective eigen vectors...

Given that the determinant of the matrix $$\left[ {\matrix{ 1 & 3 & 0 \cr 2 & 6 & 4 \cr { - 1} & 0 & 2 \cr } ...

Which one of the following describes the relationship among the three vectors, $$\widehat i + \widehat j + \widehat k,\,\,2\widehat i + 3\widehat j +...

The eigen values of a symmetric matrix are all

For the matrix $$A = \left[ {\matrix{ 5 & 3 \cr 1 & 3 \cr } } \right],$$ ONE of the normalized eigen vectors is given as

Eigen values of a real symmetric matrix are always

Consider the following system of equations $$2{x_1} + {x_2} + {x_3} = 0,\,\,{x_2} - {x_3} = 0$$ and $${x_1} + {x_2} = 0.$$ This system has

One of the eigen vector of the matrix $$A = \left[ {\matrix{ 2 & 2 \cr 1 & 3 \cr } } \right]$$ is

For a matrix $$\left[ M \right] = \left[ {\matrix{ {{3 \over 5}} & {{4 \over 5}} \cr x & {{3 \over 5}} \cr } } \right].$$ The tran...

The matrix $$\left[ {\matrix{ 1 & 2 & 4 \cr 3 & 0 & 6 \cr 1 & 1 & P \cr } } \right]$$ has one eigen value to ...

For what values of 'a' if any will the following system of equations in $$x, y$$ and $$z$$ have a solution? $$$2x+3y=4,$$$ $$$x+y+z=4,$$$ $$$x+2y-z=a$...

The number of linearly independent eigen vectors of $$\left[ {\matrix{ 2 & 1 \cr 0 & 2 \cr } } \right]$$ is

If a square matrix $$A$$ is real and symmetric then the eigen values

$$A$$ is a $$3 \times 4$$ matrix and $$AX=B$$ is an inconsistent system of equations. The highest possible rank of $$A$$ is

For what value of $$x$$ will the matrix given below become singular? $$\left[ {\matrix{ 8 & x & 0 \cr 4 & 0 & 2 \cr {12} ...

The sum of the eigen values of the matrix $$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr } } \r...

For the following set of simultaneous equations $$$1.5x - 0.5y + z = 2$$$ $$$4x + 2y + 3z = 9$$$ $$$7x + y + 5z = 10$$$

In the Gauss - elimination for a solving system of linear algebraic equations, triangularization leads to

The eigen values of $$\left[ {\matrix{ 1 & 1 & 1 \cr 1 & 1 & 1 \cr 1 & 1 & 1 \cr } } \right]$$ are

Solve the system $$2x+3y+z=9,$$ $$4x+y=7,$$ $$x-3y-7z=6$$

Among the following, the pair of vectors orthogonal to each other is

Find out the eigen value of the matrix $$A = \left[ {\matrix{ 1 & 0 & 0 \cr 2 & 3 & 1 \cr 0 & 2 & 4 \cr } } \...

The matrix $\begin{bmatrix} 1 & a \\ 8 & 3 \end{bmatrix}$ (where $a > 0$) has a negative eigenvalue if $a$ is greater than

A is a 3 × 5 real matrix of rank 2. For the set of homogeneous equations Ax = 0, where 0 is a zero vector and x is a vector of unknown variables,...

If the sum and product of eigenvalues of a 2 × 2 real matrix $\begin{bmatrix}3&p\\\ p&q\end{bmatrix} $ are 4 and -1 respec...

The system of linear equations in real (x, y) given by $\rm \begin{pmatrix} \rm x & \rm y \end{pmatrix} \begin{bmatrix} 2 & 5- 2 α \\\ α &...

Consider the matrix $$A = \left[ {\matrix{ {50} & {70} \cr {70} & {80} \cr } } \right]$$ whose eigenvectors corresponding to eigen...

The number of linear independent eigenvectors of matrix $$A = \left[ {\matrix{ 2 & 1 & 0 \cr 0 & 2 & 0 \cr 0 & 0 &...

For a given matrix $$P = \left[ {\matrix{ {4 + 3i} & { - i} \cr i & {4 - 3i} \cr } } \right],$$ where $$i = \sqrt { - 1} ,$$ the...

Choose the CORRECT set of functions, which are linearly dependent.

$$x+2y+z=4, 2x+y+2z=5, x-y+z=1$$ The system of algebraic equations given above has

The eigen vectors of the matrix $$\left[ {\matrix{ 1 & 2 \cr 0 & 2 \cr } } \right]$$ are written in the form $$\left[ {\matrix{ ...

Eigen values of a matrix $$S = \left[ {\matrix{ 3 & 2 \cr 2 & 3 \cr } } \right]$$ are $$5$$ and $$1.$$ What are the eigen values o...

Multiplication of matrices $$E$$ and $$F$$ is $$G.$$ Matrices $$E$$ and $$G$$ are $$E = \left[ {\matrix{ {\cos \theta } & { - sin\theta } &...

Which one of the following is an eigen vector of the matrix $$\left[ {\matrix{ 5 & 0 & 0 & 0 \cr 0 & 5 & 0 & 0 \cr ...