Linear Algebra · Discrete Mathematics · GATE CSE

The product of all eigenvalues of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$ is

The Lucas sequence $$L_n$$ is defined by the recurrence relation: $${L_n} = {L_{n - 1}} + {L_{n - 2}}$$, for $$n \ge 3$$, with $${L_1} = 1$$ and $${L_...

Let $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 \cr 4 & 1 & 2 & 3 \cr 3 & 4 & 1 & 2 \cr 2 & 3 & 4 & 1 \cr } } \right]$$ and $$B = \left...

Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}. Let $$\lambda_1,\lambda_2,\lambda_3,\lambda_4$$, and $$\lambda_5$$ be the f...

Consider the following two statements with respect to the matrices Am $$\times$$ n , Bn $$\times$$ m , Cn$$\times$$ n and Dn $$\times$$ n . Statement...

Suppose that P is a 4 × 5 matrix such that every solution of the equation Px = 0 is a scalar multiple of [2 5 4 3 1]T​. The rank of P is ___...

Let X be a square matrix. Consider the following two statements on X. I. X is invertible. II. Determinant of X is non-zero. Which one of the followin...

Consider a matrix $$A = u{v^T}$$ where $$u = \left( {\matrix{ 1 \cr 2 \cr } } \right),v = \left( {\matrix{ 1 \cr 1 \cr } } \ri...

Let $$P = \left[ {\matrix{ 1 & 1 & { - 1} \cr 2 & { - 3} & 4 \cr 3 & { - 2} & 3 \cr } } \right]$$ and $$Q = \...

Let $${c_1},.....,\,\,{c_n}$$ be scalars, not all zero, such that $$\sum\limits_{i = 1}^n {{c_i}{a_i} = 0} $$ where $${{a_i}}$$ are column vectors in ...

Suppose that the eigen values of matrix $$A$$ are $$1, 2, 4.$$ The determinant of $${\left( {{A^{ - 1}}} \right)^T}$$ is _______.

Consider the system, each consisting of m linear equations in $$n$$ variables. $$I.$$ $$\,\,\,$$ If $$m < n,$$ then all such system have a solutio...

Two eigenvalues of a $$3 \times 3$$ real matrix $$P$$ are $$\left( {2 + \sqrt { - 1} } \right)$$ and $$3.$$ The determinant of $$P$$ is _______.

Let $${a_n}$$ be the number of $$n$$-bit strings that do NOT contain two consecutive $$1s.$$ Which one of the following is the recurrence relation for...

In the LU decomposition of the matrix $$\left[ {\matrix{ 2 & 2 \cr 4 & 9 \cr } } \right]$$, if the diagonal elements of U are both...

In the given matrix $$\left[ {\matrix{ 1 & { - 1} & 2 \cr 0 & 1 & 0 \cr 1 & 2 & 1 \cr } } \right],$$ one of t...

The number of divisors of $$2100$$ is ___________.

The larger of the two eigenvalues of the matrix $$\left[ {\matrix{ 4 & 5 \cr 2 & 1 \cr } } \right]$$ is ______.

If the matrix A is such that $$$A = \left[ {\matrix{ 2 \cr { - 4} \cr 7 \cr } } \right]\,\,\left[ {\matrix{ 1 & 9 & 5 \c...

Which one of the following statements is TRUE about every $$n\,\, \times \,n$$ matrix with only real eigen values?

If $${V_1}$$ and $${V_2}$$ are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of $${V_1}\, \cap \,\,{...

The value of the dot product of the eigenvectors corresponding to any pair of different eigen values of a 4-by-4 symmetric positive definite matrix is...

Consider the following system of equations: 3x + 2y = 1 4x + 7z = 1 x + y + z =3 x - 2y + 7z = 0 The number of solutions for this system is _____...

Which of the following does not equal $$\left| {\matrix{ 1 & x & {{x^2}} \cr 1 & y & {{y^2}} \cr 1 & z & {{z^2}}...

Let $$A$$ be the $$2 \times 2$$ matrix with elements $${a_{11}} = {a_{12}} = {a_{21}} = + 1$$ and $${a_{22}} = - 1$$. Then the eigen values of the m...

Consider the following matrix $$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right].$$ If the eigen values of $$A$$ are $$4$$ an...

The following system of equations $${x_1}\, + \,{x_2}\, + 2{x_3}\, = 1$$ $${x_1}\, + \,2 {x_2}\, + 3{x_3}\, = 2$$ $${x_1}\, + \,4{x_2}\, + a{x_3}\,...

Let $$A$$ be the matrix $$\left[ {\matrix{ 3 & 1 \cr 1 & 2 \cr } } \right]$$. What is the maximum value of $${x^T}Ax$$ where the ...

The determination of the matrix given below is $$$\left[ {\matrix{ 0 & 1 & 0 & 2 \cr { - 1} & 1 & 1 & 3 \cr 0 &am...

The number of different $$n \times n$$ symmetric matrices with each elements being either $$0$$ or $$1$$ is

What values of x, y and z satisfy the following system of linear equations? $$$\left[ {\matrix{ 1 & 2 & 3 \cr 1 & 3 & 4 \cr ...

Let A, B, C, D be $$n\,\, \times \,\,n$$ matrices, each with non-zero determination. If ABCD = I, then $${B^{ - 1}}$$ is

$$A$$ system of equations represented by $$AX=0$$ where $$X$$ is a column vector of unknown and $$A$$ is a square matrix containing coefficients has a...

The rank of the matrix$$\left[ {\matrix{ 1 & 1 \cr 0 & 0 \cr } } \right]\,\,is$$

Consider the following statements: S1: The sum of two singular n x n matrices may be non-singular S2: The sum of two n x n non-singular matrices may...

An $$n\,\, \times \,\,n$$ array v is defined as follows v[i, j] = i - j for all i, j, $$1\,\, \le \,\,i\,\, \le \,\,n,\,1\,\, \le \,\,j\,\, \le \,\,n$...

The determinant of the matrix $$$\left[ {\matrix{ 2 & 0 & 0 & 0 \cr 8 & 1 & 7 & 2 \cr 2 & 0 & 2 & 0 ...

Consider the following set a equations x + 2y = 5 4x + 8y = 12 3x + 6y + 3z = 15 This set

The determination of the matrix $$$\left[ {\matrix{ 6 & { - 8} & 1 & 1 \cr 0 & 2 & 4 & 6 \cr 0 & 0 & 4 &a...

Let AX = B be a system of linear equations where A is an m x n matrix and B is a $$m\,\, \times \,\,1$$ column vector and X is a n x 1 column vector o...

Let $$A = \left[ {\matrix{ {{a_{11}}} & {{a_{12}}} \cr {{a_{21}}} & {{a_{22}}} \cr } } \right]\,\,$$ and $$B = \left[ {\matrix{ ...

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

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

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

The rank of the matrix $$\left[ {\matrix{ 0 & 0 & { - 3} \cr 9 & 3 & 5 \cr 3 & 1 & 1 \cr } } \right]$$ is

The eigen vector (s) of the matrix $$\left[ {\matrix{ 0 & 0 & \alpha \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \r...

If $$A = \left[ {\matrix{ 1 & 0 & 0 & 1 \cr 0 & { - 1} & 0 & { - 1} \cr 0 & 0 & i & i \cr 0 &...

Let A be an n × n matrix over the set of all real numbers ℝ. Let B be a matrix obtained from A by swapping two rows. Which of the following statements...

Let A be any n x m matrix, where m > n. Which of the following statements is/are TRUE about the system of linear equations Ax = 0?...

Which one of the following is the closed form for the generating function of the sequence (an}n $$\ge$$ 0 defined below? $${a_n} = \left\{ {\matrix{ ...

Consider solving the following system of simultaneous equations using LU decomposition. x1 + x2 $$-$$ 2x3 = 4 x1 + 3x2 $$-$$ x3 = 7 2x1 + x2 $$-$$ 5x3...

Which of the following is/are the eigenvector(s) for the matrix given below? $$\left( {\matrix{ { - 9} & { - 6} & { - 2} & { - 4} \cr { - 8} &...

For two n-dimensional real vectors P and Q, the operation s(P, Q) is defined as follows: $$s\left( {P,\;Q} \right) = \mathop \sum \limits_{i = 1}^n \...

Consider the following matrix. $$\left( {\begin{array}{*{20}{c}} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&...

Let A and B be two n$$ \times $$n matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Cons...

Consider a matrix P whose only eigenvectors are the multiples of $$\left[ {\matrix{ 1 \cr 4 \cr } } \right].$$ Consider the following sta...

Which one of the following is a closed form expression for the generating function of the sequence $$\left\{ {{a_n}} \right\},$$ where $${a_n} = 2n + ...

If the characteristic polynomial of a $$3 \times 3$$ matrix $$M$$ over $$R$$(the set of real numbers) is $${\lambda ^3} - 4{\lambda ^2} + a\lambda + ...

Let $$A$$ be $$n\,\, \times \,\,n$$ real valued square symmetric matrix of rank $$2$$ with $$\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {A_{ij}^2 = ...

The value of the expression $${13^{99}}$$ ($$mod$$ $$17$$), in the range $$0$$ to $$16,$$ is ______________ .

Let $${A_1},\,{A_2},\,{A_3}$$ and $${A_4}$$ be four matrices of dimensions $$10 \times 5,\,5 \times 20,\,20 \times 10,$$ and $$10 \times 5,$$ respecti...

Consider the recurrence relation $${a_1} = 8,\,{a_n} = 6{n^2} + 2n + {a_{n - 1}}.$$ Let $${a_{99}} = K \times {10^4}.$$ The value of $$K$$ is _______...

Let $${a_n}$$ represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence relation for $${a_n}$$?

$$\sum\limits_{x = 1}^{99} {{1 \over {x\left( {x + 1} \right)}}} $$ = _____________.

Consider the following $$2 \times 2$$ matrix $$A$$ where two elements are unknown and are marked by $$a$$ and $$b.$$ The eigenvalues of this matrix a...

If the following system has non - trivial solution $$$px+qy+rz=0$$$ $$$qx+ry+pz=0$$$ $$$rx+py+qz=0$$$ Then which one of the following Options is TRUE...

Perform the following operations on the matrix $$\left[ {\matrix{ 3 & 4 & {45} \cr 7 & 9 & {105} \cr {13} & 2 & {...

The product of the non-zero eigenvalues of the matrix $$\left[ {\matrix{ 1 & 0 & 0 & 0 & 1 \cr 0 & 1 & 1 & 1 &...

Four matrices $${M_1},\,\,\,{M_2},\,\,\,{M_3}$$ and $${M_4}$$ of dimensions $$p\,\,x\,\,q,\,\,\,\,\,q\,\,x\,\,e,\,\,\,\,\,r\,\,x\,\,s$$ and $$\,\,\,\,...

Consider the matrix as given below. $$$\left[ {\matrix{ 1 & 2 & 3 \cr 0 & 4 & 7 \cr 0 & 0 & 3 \cr } } \right...

$$\left[ A \right]$$ is a square matrix which is neither symmetric nor skew-symmetric and $${\left[ A \right]^T}$$ is its transpose. The sum and diffe...

Consider the following matrix $$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right]\,\,$$ If the eigen values of $$A$$ are $$4$$ ...

If $$M$$ is a square matrix with a zero determinant, which of the following assertion(s) is (are) correct? $$S1$$ : Each row of $$M$$ can be represen...

How many of the following matrices have an eigen value $$1$$? $$\left[ {\matrix{ 1 & 0 \cr 0 & 0 \cr } } \right],\,\,\left[ {\matr...

Let $$A$$ be $$a$$ $$4$$ $$x$$ $$4$$ matrix with eigen values $$-5$$, $$-2, 1, 4$$. Which of the following is an eigen value of $$\left[ {\matrix{ ...

$$F$$ is an $$n$$ $$x$$ $$n$$ real matrix. $$b$$ is an $$n$$ $$x$$ $$1$$ real vector. Suppose there are two $$n$$ $$x$$ $$1$$ vectors, $$u$$ and $$v$$...

What are the eigen values of the matrix $$P$$ given below? $$$P = \left( {\matrix{ a & 1 & 0 \cr 1 & a & 1 \cr 0 & 1 ...

Consider the set $$H$$ of all $$3$$ $$X$$ $$3$$ matrices of the type $$$\left[ {\matrix{ a & f & e \cr 0 & b & d \cr 0 &a...

What are the eigen values of the following $$2x2$$ matrix? $$$\left[ {\matrix{ 2 & { - 1} \cr { - 4} & 5 \cr } } \right]$$$

Consider the following system of equations in three real variables $$x1, x2$$ and $$x3$$ : $$2x1 - x2 + 3x3 = 1$$ $$3x1 + 2x2 + 5x3 = 2$$ $$ - x1 + 4x...

In an M$$ \times $$N matrix such that all non-zero entries are covered in $$a$$ rows and $$b$$ columns. Then the maximum number of non-zero entries, s...

If matrix $$X = \left[ {\matrix{ a & 1 \cr { - {a^2} + a - 1} & {1 - a} \cr } } \right]$$ and $${X^2} - X + 1 = 0$$ ($${\rm I}$$ ...

Let $$A$$ be and n$$ \times $$n matrix of the folowing form. What is the value of the determinant of $$A$$?...

How many solutions does the following system of linear equations have? - x + 5y = - 1x - y = 2x + 3y = 3

Consider the following system of linear equations $$$\left[ {\matrix{ 2 & 1 & { - 4} \cr 4 & 3 & { - 12} \cr 1 & 2 &...

Obtain the eigen values of the matrix $$$A = \left[ {\matrix{ 1 & 2 & {34} & {49} \cr 0 & 2 & {43} & {94} \cr 0 &...

Consider the following determinant $$$\Delta = \left| {\matrix{ 1 & a & {bc} \cr 1 & a & {ca} \cr 1 & a & {ab} ...

The rank of the matrix given below is: $$$\left[ {\matrix{ 1 & 4 & 8 & 7 \cr 0 & 0 & 3 & 0 \cr 4 & 2 & 3 ...

Let $$A = ({a_{ij}})$$ be and n-rowed square matrix and $${I_{12}}$$ be the matrix obtained by interchanging the first and second rows of the n-rowed ...

The matrices$$\left[ {\matrix{ {\cos \,\theta } & { - \sin \,\theta } \cr {\sin \,\,\theta } & {\cos \,\,\theta } \cr } } \right]\...

If A and B are real symmetric matrices of size n x n. Then, which one of the following is true?

In a compact single dimensional array representation for lower triangular matrices (i.e., all the elements above the diagonal are zero) of size $$n$$ ...

A square matrix is singular whenever:

If a, b and c are constants, which of the following is a linear inequality?