Linear Algebra

Consider the system of linear equations given below. $$ \begin{aligned} a x+y & =b \\ 16 x+a y & =24 \end{aligned} $$ Suppose the values of a and b are chosen such that the system of linear equations produce multiple sol...

For $n>1$, the maximum multiplicity of any eigenvalue of an $n \times n$ matrix with elements from $\mathbb{R}$ is

A student needs to enroll for a minimum of 60 credits. A student cannot enroll for more than 70 credits. The credits are divided amongst project and three distinct sets of courses namely, core courses, specialization cou...

Consider $4 \times 4$ matrices with their elements from $\{0,1\}$. The number of such matrices with even number of 1 s in every row and every column is

Let $n>1$. Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$. Let the vector ( 0,1 , $0,0, \ldots, 0) \in \mathbb{R}^n$ be in the null space of $M$. Which of the following options is/are always corr...

If $A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)$, then which ONE of the following is $A^8$ ?

Consider the given system of linear equations for variables $x$ and $y$, where $k$ is a realvalued constant. Which of the following option(s) is/are CORRECT? $$\begin{aligned} & x+k y=1 \\ & k x+y=-1 \end{aligned}$$

Consider a system of linear equations $P X=Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 3}$. Suppose $P$ has an $L U$ decomposition, $P=L U$, where $$L=\left[\begin{array}{ccc} 1 & 0 & 0 \\ l_...

Let $L, M$, and $N$ be non-singular matrices of order 3 satisfying the equations $L^2=L^{-1}, M=L^8$ and $N=L^2$. Which ONE of the following is the value of the determinant of $(M-N)$ ?

The average marks obtained by a class in an examination were calculated as 30.8 . However, while checking the marks entered, the teacher found that the marks of one student were entered incorrectly as 24 instead of 42 ....

Let $A$ be a $2 \times 2$ matrix as given. $$A=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$$ What are the eigenvalues of the matrix $A^{13}$ ?

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 ?

If two distinct non-zero real variables $x$ and $y$ are such that $(x + y)$ is proportional to $(x - y)$ then the value of $\frac{x}{y}$

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 is/are TRUE?

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[ {\matrix{ 3 & 4 & 1 & 2 \cr 4 & 1 & 2 & 3 \cr 1 & 2 & 3 & 4 \cr 2 & 3 & 4 & 1 \cr } } \rig...

Consider solving the following system of simultaneous equations using LU decomposition. x 1 + x 2 $$-$$ 2x 3 = 4 x 1 + 3x 2 $$-$$ x 3 = 7 2x 1 + x 2 $$-$$ 5x 3 = 7 where L and U are denoted as $$L = \left( {\matrix{ {{L_...

Which of the following is/are the eigenvector(s) for the matrix given below? $$\left( {\matrix{ { - 9} & { - 6} & { - 2} & { - 4} \cr { - 8} & { - 6} & { - 3} & { - 1} \cr {20} & {15} & 8 & 5 \cr {32} & {21} & 7 & {12} \...

Consider the following two statements with respect to the matrices A m $$\times$$ n , B n $$\times$$ m , C n$$\times$$ n and D n $$\times$$ n . Statement 1 : tr(AB) = tr(BA) Statement 2 : tr(CD) = tr(DC) where tr( ) repr...

Consider the following matrix. $$\left( {\begin{array}{*{20}{c}} 0&1&1&1\\ 1&0&1&1\\ 1&1&0&1\\ 1&1&1&0 \end{array}} \right)$$ The largest eigenvalue of the above matrix is ______

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

The number of student in three classes is in the ratio 3 : 13 : 6. If 18 students are added to each class, the ratio changes to 15 : 35 : 21. The total number of students in all the three classes in the beginning was :

If $$\theta$$ is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then cos $$\theta$$ = _______

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

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. Consider the following statements, I. rank(AB) = rank(A) rank(B) II. det(A...

Ten friends planned to share equally the cost of buying a gift for their teacher. When two of them decided not to contribute, each of the other friends had to pay Rs 150 more. The cost of the gift was Rs. ___________.

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 following is TRUE?

Consider the following matrix : $$ R=\left[\begin{array}{cccc} 1 & 2 & 4 & 8 \\ 1 & 3 & 9 & 27 \\ 1 & 4 & 16 & 64 \\ 1 & 5 & 25 & 125 \end{array}\right] $$ The absolute value of the product of Eigen values of $R$ is ____...

Consider a matrix $$A = u{v^T}$$ where $$u = \left( {\matrix{ 1 \cr 2 \cr } } \right),v = \left( {\matrix{ 1 \cr 1 \cr } } \right).$$ Note that $${v^T}$$ denotes the transpose of $$v.$$ The largest eigenvalue of $$A$$ is...

Consider a matrix P whose only eigenvectors are the multiples of $$\left[ {\matrix{ 1 \cr 4 \cr } } \right].$$ Consider the following statements. $$\left( {\rm I} \right)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$P$$ does not...

In appreciation of the social improvements completed in a town, a wealthy philanthropist decided to gift Rs $$750$$ to each male senior citizen in the town and Rs $$1000$$ to each female senior citizen. Altogether, there...

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 $${R^{11}}.$$ Consider the set of linear equations $$AX=b$$ Where $$A...

Let $$P = \left[ {\matrix{ 1 & 1 & { - 1} \cr 2 & { - 3} & 4 \cr 3 & { - 2} & 3 \cr } } \right]$$ and $$Q = \left[ {\matrix{ { - 1} & { - 2} & { - 1} \cr 6 & {12} & 6 \cr 5 & {10} & 5 \cr } } \right]$$ be two matrices. T...

If the characteristic polynomial of a $$3 \times 3$$ matrix $$M$$ over $$R$$(the set of real numbers) is $${\lambda ^3} - 4{\lambda ^2} + a\lambda + 30.\,a \in R,$$ and one eigenvalue of $$M$$ is $$2,$$ then the largest...

Let $$A$$ be $$n\,\, \times \,\,n$$ real valued square symmetric matrix of rank $$2$$ with $$\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {A_{ij}^2 = 50.} } $$ Consider the following statements. $$(I)$$ One eigenvalue mu...

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 solution $$II.$$ $$\,\,\,$$ If $$m > n,$$ then none of these systems has a soluti...

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

A function $$f(x)$$ is linear and has a value of $$29$$ at $$x=-2$$ and $$39$$ at $$x=3.$$ Find its value at $$x=5.$$

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 $$\left[ {\matrix{ 3 & 4 & {45} \cr 7 & 9 & {105} \cr {13} & 2 & {195} \cr } } \right]$$ (i) Add the third row to the second row (ii) Subtract the third column from the firs...

In the given matrix $$\left[ {\matrix{ 1 & { - 1} & 2 \cr 0 & 1 & 0 \cr 1 & 2 & 1 \cr } } \right],$$ one of the eigenvalues is $$1.$$ The eigen vectors corresponding to the eigen value $$1$$ are

In the LU decomposition of the matrix $$\left[ {\matrix{ 2 & 2 \cr 4 & 9 \cr } } \right]$$, if the diagonal elements of U are both 1, then the lower diagonal entry $${l_{22}}$$ of L is ________.

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 ar $$-1$$ and $$7.$$ What are the values of $$a$$ and $$b$$? $$A = \left...

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 $$$A = \left[ {\matrix{ 2 \cr { - 4} \cr 7 \cr } } \right]\,\,\left[ {\matrix{ 1 & 9 & 5 \cr } } \right]$$$ then the determinant of A is equal to _________.

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

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 ______________________

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

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

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 matrix $${A^{19}}$$ are

$$\left[ A \right]$$ is a square matrix which is neither symmetric nor skew-symmetric and $${\left[ A \right]^T}$$ is its transpose. The sum and differences of these matrices and defined as $$\left[ S \right] = \left[ A...

Consider the matrix as given below. $$$\left[ {\matrix{ 1 & 2 & 3 \cr 0 & 4 & 7 \cr 0 & 0 & 3 \cr } } \right]$$$ Which of the following options provides the Correct values of the Eigen values of the matrix?

Consider the following matrix $$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right].$$ If the eigen values of $$A$$ are $$4$$ and $$8$$ then

Consider the following matrix $$A = \left[ {\matrix{ 2 & 3 \cr x & y \cr } } \right]\,\,$$ If the eigen values of $$A$$ are $$4$$ and $$8$$, then

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 represented as a linear combination of the other rows $$S2$$ : Each column of $...

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

The following system of equations $${x_1}\, + \,{x_2}\, + 2{x_3}\, = 1$$ $${x_1}\, + \,2 {x_2}\, + 3{x_3}\, = 2$$ $${x_1}\, + \,4{x_2}\, + a{x_3}\, = 4$$ has a unique solution. The only possible value (s) for $$\alpha $$...

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 maximum is taken over all $$x$$ that are the unit eigenvectors of $$A$$?

Consider the set of (column) vectors defined by $$X = \,\{ \,x\, \in \,{R^3}\,\left| {{x_1}\, + \,{x_2}\, + \,{x_3} = 0} \right.$$, where $${x^T} = \,{[{x_1}\, + \,{x_2}\, + \,{x_3}]^T}\} .$$ Which of the following is TR...

Let $$A$$ be $$a$$ $$4$$ $$x$$ $$4$$ matrix with eigen values $$-5$$, $$-2, 1, 4$$. Which of the following is an eigen value of $$\left[ {\matrix{ {\rm A} & {\rm I} \cr {\rm I} & {\rm A} \cr } } \right]$$, where $$I$$ is...

$$F$$ 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 $$v$$ such that $$u \ne v$$, and $$Fu = b,\,\,\,\,Fv = b$$ Which one of the...

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

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

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

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

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

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

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

Consider the following system of linear equations $$$\left[ {\matrix{ 2 & 1 & { - 4} \cr 4 & 3 & { - 12} \cr 1 & 2 & { - 8} \cr } } \right]\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ \alpha \cr 5 \c...

$$A$$ 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 non-trival solution when $$A$$ is.

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

Obtain the eigen values of the matrix $$$A = \left[ {\matrix{ 1 & 2 & {34} & {49} \cr 0 & 2 & {43} & {94} \cr 0 & 0 & { - 2} & {104} \cr 0 & 0 & 0 & { - 1} \cr } } \right]$$$

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 be singular Which of the following statements is correct?

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 sum of elements of the array v is

The determinant of the matrix $$$\left[ {\matrix{ 2 & 0 & 0 & 0 \cr 8 & 1 & 7 & 2 \cr 2 & 0 & 2 & 0 \cr 9 & 0 & 6 & 1 \cr } } \right]\,\,is$$$

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

Consider the following determinant $$$\Delta = \left| {\matrix{ 1 & a & {bc} \cr 1 & a & {ca} \cr 1 & a & {ab} \cr } } \right|$$$ Which of the following is a factor of $$\Delta $$ ?

The rank of the matrix given below is: $$$\left[ {\matrix{ 1 & 4 & 8 & 7 \cr 0 & 0 & 3 & 0 \cr 4 & 2 & 3 & 1 \cr 3 & {12} & {24} & {2} \cr } } \right]$$$

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 Identity matrix. Then$${AI_{12}}$$ is such that its first

The determination of the matrix $$$\left[ {\matrix{ 6 & { - 8} & 1 & 1 \cr 0 & 2 & 4 & 6 \cr 0 & 0 & 4 & 8 \cr 0 & 0 & 0 & { - 1} \cr } } \right]\,\,is$$$

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

The matrices$$\left[ {\matrix{ {\cos \,\theta } & { - \sin \,\theta } \cr {\sin \,\,\theta } & {\cos \,\,\theta } \cr } } \right]\,\,and$$ $$\left[ {\matrix{ a & 0 \cr 0 & b \cr } } \right]\,$$ commute under multiplicati...

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 of unknowns. Which of the following is false?

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}} & . & . & . & {{a^n}} \cr . & . & . & . & . & . & . \cr . & ....

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}} & . & . & . & {{a^n}} \cr . & . & . & . & . & . & . \cr . & ....

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

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

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

If $$A = \left[ {\matrix{ 1 & 0 & 0 & 1 \cr 0 & { - 1} & 0 & { - 1} \cr 0 & 0 & i & i \cr 0 & 0 & 0 & { - i} \cr } } \right]$$ the matrix $${A^4},$$ calculated by the use of Cayley - Hamilton theoram (or) otherwise is

The eigen vector (s) of the matrix $$\left[ {\matrix{ 0 & 0 & \alpha \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right],\alpha \ne 0$$ is (are)

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

A square matrix is singular whenever: