determinant

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 𝑥 2 + 𝑦 2 ≤ 1 gets transformed to a...

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 & 1 \end{bmatrix} = \rm \begin{pmatrix} \rm...

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 respectively, then |p| is _______ (in integer).

For three vectors $$\vec A = 2\hat j - 3\hat k,\vec B = - 2\hat i + \hat k\ and\;\vec C = 3\hat i - \hat j,$$ where î, ĵ and k̂ are unit vectors along the axes of a right-handed re...

The set of equations x + y + z = 1 ax - ay + 3z = 5 5x - 3y + az = 6 has infinite solutions, if a =

The determinant of a 2×2 matrix is 50. If one eigenvalue of the matrix is 10, the other eigenvalue is

If A = $\begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 0 & 0 & 1 \end{bmatrix}$ then det(A$^{-1}$) is ________ (correct to two decimal places).

Consider the matrix P = $\begin{bmatrix} \frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\ 0 & 1 & 0 \\ -\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \end{bmatrix}$. Which one of the f...

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

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

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

If any two columns of a determinant $$P = \left| {\matrix{ 4 & 7 & 8 \cr 3 & 1 & 5 \cr 9 & 6 & 2 \cr } } \right|$$ are interchanged, which one of the following statements regarding...

At least one eigenvalue of a singular matrix is

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

Which one of the following describes the relationship among the three vectors, $$\widehat i + \widehat j + \widehat k,\,\,2\widehat i + 3\widehat j + \widehat k$$ and $$5\widehat i...

Given that the determinant of the matrix $$\left[ {\matrix{ 1 & 3 & 0 \cr 2 & 6 & 4 \cr { - 1} & 0 & 2 \cr } } \right]$$ is $$-12$$, the determinant of the matrix $$\left[ {\matrix...

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

For what value of $$x$$ will the matrix given below become singular? $$\left[ {\matrix{ 8 & x & 0 \cr 4 & 0 & 2 \cr {12} & 6 & 0 \cr } } \right]$$

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