linear algebra

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 $...

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...

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: $Ax = b$, where $A$ is an $n \times n$ matrix, and $x$ and $b$ are $n$-dimensional column vectors. Suppose this system of equations has a u...

Let v₁ and v₂ be the two eigenvectors corresponding to distinct eigenvalues of a 3 × 3 real symmetric matrix. Which one of the following statements is true?

Let $A = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1 \end{bmatrix}$ and $b = \begin{bmatrix} 1/3 \\ -1/3 \\ 0 \end{bmatrix}$. Then, the system of linear equations $Ax =...

Let $v_1$ and $v_2$ be the two eigen vectors corresponding to distinct eigen values of a $3 \times 3$ real symmetric matrix. Which one of the following statements is true?

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 l...

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).

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 w = [1 2 3]$^T$, 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

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

e 4 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:...

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

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 $\_\_\_\_$ .

M is a 2 x 2 matrix with eigenvalues 4 and 9. The eigenvalues of M² are

The rank of the matrix, M = [0 1 1] [1 0 1], [1 1 0] is _________.

The matrix A = $\begin{bmatrix} 3/2 & 0 & 1/2 \ 0 & -1 & 0 \ 1/2 & 0 & 3/2 \end{bmatrix}$ has three distinct eigenvalues and one of its eigenvectors is $\begin{bmatrix} 1 \ 0 \ 1 \...

Consider a non-singular $2 \times 2$ square matrix $\mathbf{A}$. If $trace(\mathbf{A})=4$ and $trace(\mathbf{A}^2)=5$, the determinant of the matrix $\mathbf{A}$ is ________ (up to...

The eigenvalues of the matrix given below are $$\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -3 & -4 \end{bmatrix}$$

Let A = $\begin{bmatrix} 1 & 0 & -1 \\ -1 & 2 & 0 \\ 0 & 0 & -2 \end{bmatrix}$ and $B = A^3 - A^2 - 4A + 5I$, where $I$ is the $3 \times 3$ identity matrix. The determinant of $B$...

The matrix $$A = \left[ {\matrix{ {{3 \over 2}} & 0 & {{1 \over 2}} \cr 0 & { - 1} & 0 \cr {{1 \over 2}} & 0 & {{3 \over 2}} \cr } } \right]$$ has three distinct eigen values and o...

The eigen values of the matrix given below are $$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & { - 3} & { - 4} \cr } } \right]$$

Consider $$3 \times 3$$ matrix with every element being equal to $$1.$$ Its only non-zero eigenvalue is __________.

Let the eigenvalues of a $$2 \times 2$$ matrix $$A$$ be $$1,-2$$ with eigenvectors $${x_1}$$ and $${x_2}$$ respectively. Then the eigenvalues and eigenvectors of the matrix $${A^2}...

$$A$$ $$3 \times 3$$ matrix $$P$$ is such that , $${p^3} = P.$$ Then the eigen values of $$P$$ are

Let $$P = \left[ {\matrix{ 3 & 1 \cr 1 & 3 \cr } } \right].$$ Consider the set $$S$$ of all vectors $$\left( {\matrix{ x \cr y \cr } } \right)$$ such that $${a^2} + {b^2} = 1$$ whe...

We have a set of $$3$$ linear equations in $$3$$ unknown. $$'X \equiv Y'$$ means $$X$$ and $$Y$$ are equivalent statements and $$'X \ne Y'$$ means $$X$$ and $$y$$ are not equivalen...

Given a system of equations $$$x + 2y + 2z = {b_1}$$$ $$$5x + y + 3z = {b_2}$$$ Which of the following is true its solutions

Which one of the following statements is true for all real symmetric matrices?

A system matrix is given as follows $$$A = \left[ {\matrix{ 0 & 1 & { - 1} \cr { - 6} & { - 11} & 6 \cr { - 6} & { - 11} & 5 \cr } } \right].$$$ The absolute value of the ratio of...

$$A = \left[ {\matrix{ p & q \cr r & s \cr } } \right];B = \left[ {\matrix{ {{p^2} + {q^2}} & {pr + qs} \cr {pr + qs} & {{r^2} + {s^2}} \cr } } \right]$$ If the rank of matrix $$A$...

The equation $$\left[ {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr } } \right]$$ has

The two vectors $$\left[ {\matrix{ 1 & 1 & 1 \cr } } \right]$$ and $$\left[ {\matrix{ 1 & a & {{a^2}} \cr } } \right]$$ where $$a = - {1 \over 2} + j{{\sqrt 3 } \over 2}$$ and $$j...

The matrix $$\left[ A \right] = \left[ {\matrix{ 2 & 1 \cr 4 & { - 1} \cr } } \right]$$ is decomposed into a product of lower triangular matrix $$\left[ L \right]$$ and an upper tr...

For the set of equations $$${x_1} + 2{x_2} + {x_3} + 4{x_4} = 2,$$$ $$$3{x_1} + 6{x_2} + 3{x_3} + 12{x_4} = 6.$$$ The following statement is true

An eigen vector of $$p = \left[ {\matrix{ 1 & 1 & 0 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$ is

The trace and determinant of a $$2 \times 2$$ matrix are shown to be $$-2$$ and $$-35$$ respectively. Its eigen values are

$$A$$ is $$m$$ $$x$$ $$n$$ full rank matrix with $$m > n$$ and $${\rm I}$$ is an identity matrix. Let matrix $${A^ + } = {\left( {{A^T}A} \right)^{ - 1}}{A^T}.$$ Then which one of...

Let $$P$$ be $$2x2$$ real orthogonal matrix and $$\overline x $$ is a real vector $${\left[ {\matrix{ {{x_1}} & {{x_2}} \cr } } \right]^T}$$ with length $$\left| {\left| {\overline...

The characteristic equation of a $$3\,\, \times \,\,3$$ matrix $$P$$ is defined as $$\alpha \left( \lambda \right) = \left| {\lambda {\rm I} - P} \right| = {\lambda ^3} + 2\lambda...

If the rank of a $$5x6$$ matrix $$Q$$ is $$4$$ then which one of the following statements is correct?

Let $$x$$ and $$y$$ be two vectors in a $$3-$$ dimensional space and $$ < x,y > $$ denote their dot product. Then the determinant det $$\left[ {\matrix{ { < x,x > } & { < x,y > } \...

If $$A = \left[ {\matrix{ { - 3} & 2 \cr { - 1} & 0 \cr } } \right]\,$$ then $${A^9}$$ equals

$$X = {\left[ {\matrix{ {{x_1}} & {{x_2}} & {.......\,{x_n}} \cr } } \right]^T}$$ is an $$n$$-tuple non- zero vector. The $$n\,\, \times \,\,n$$ matrix $$V = X{X^T}$$

For the matrix $$P = \left[ {\matrix{ 3 & { - 2} & 2 \cr 0 & { - 2} & 1 \cr 0 & 0 & 1 \cr } } \right],$$ one of the eigen values is $$-2.$$ Which of the following is an eigen vecto...

If $$R = \left[ {\matrix{ 1 & 0 & { - 1} \cr 2 & 1 & { - 1} \cr 2 & 3 & 2 \cr } } \right]$$ then the top row of $${R^{ - 1}}$$ is

In the matrix equation $$PX=Q$$ which of the following is a necessary condition for the existence of atleast one solution for the unknown vector $$X.$$

The determinant of the matrix $$\left[ {\matrix{ 1 & 0 & 0 & 0 \cr {100} & 1 & 0 & 0 \cr {100} & {200} & 1 & 0 \cr {100} & {200} & {300} & 1 \cr } } \right]$$ is

If $$A = \left[ {\matrix{ 1 & { - 2} & { - 1} \cr 2 & 3 & 1 \cr 0 & 5 & { - 2} \cr } } \right]$$ and $$adj (A)$$ $$ = \left[ {\matrix{ { - 11} & { - 9} & 1 \cr 4 & { - 2} & { - 3}...

Find the eigen values and eigen vectors of the matrix $$\left[ {\matrix{ 3 & { - 1} \cr { - 1} & 3 \cr } } \right]$$

If $$A = \left[ {\matrix{ 5 & 0 & 2 \cr 0 & 3 & 0 \cr 2 & 0 & 1 \cr } } \right]$$ then $${A^{ - 1}} = $$

A set of linear equations is represented by the matrix equations $$Ax=b.$$ The necessary condition for the existence of a solution for this system is

$$A = \left[ {\matrix{ 2 & 0 & 0 & { - 1} \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 3 & 0 \cr { - 1} & 0 & 0 & 4 \cr } } \right].$$ The sum of the eigen values of the matrix $$A$$ is

If the vector $$\left[ {\matrix{ 1 \cr 2 \cr { - 1} \cr } } \right]$$ is an eigen vector of $$A = \left[ {\matrix{ { - 2} & 2 & { - 3} \cr 2 & 1 & { - 6} \cr { - 1} & { - 2} & 0 \c...

Express the given matrix $$A = \left[ {\matrix{ 2 & 1 & 5 \cr 4 & 8 & {13} \cr 6 & {27} & {31} \cr } } \right]$$ as a product of triangular matrices $$L$$ and $$U$$ where the diago...

The inverse of the matrix $$S = \left[ {\matrix{ 1 & { - 1} & 0 \cr 1 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]$$ is

The rank of the following $$(n+1)$$ $$x$$ $$(n+1)$$ matrix, where $$'a'$$ is a real number is $$$\left[ {\matrix{ 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr 1 & a & {{a^2}} & . & ....

Given the matrix $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr { - 6} & { - 11} & { - 6} \cr } } \right].\,\,$$ Its eigen values are

$$A$$ $$\,\,5 \times 7$$ matrix has all its entries equal to $$1.$$ Then the rank of a matrix is

The number of linearly independent solutions of the system of equations $$\left[ {\matrix{ 1 & 0 & 2 \cr 1 & { - 1} & 0 \cr 2 & { - 2} & 0 \cr } } \right]\,\,\left[ {\matrix{ {{x_1...

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