Linear Algebra (CE)

Suppose $\lambda$ is an eigenvalue of matrix A and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $\mathrm{B}=\mathrm{A}-2 \mathrm{I}$, where I is the identity matrix. Then, the eigenv...

For the matrix $[\mathrm{A}]$ given below, the transpose is $\qquad$ . $$ [A]=\left[\begin{array}{lll} 2 & 3 & 4 \\ 1 & 4 & 5 \\ 4 & 3 & 2 \end{array}\right] $$

Let $A=\left[\begin{array}{cc}1 & 1 \\ 1 & 3 \\ -2 & -3\end{array}\right]$ and $b=\left[\begin{array}{l}b_1 \\ b_2 \\ b_3\end{array}\right]$. For $\mathrm{Ax}=\mathrm{b}$ to be solvable, which one of the following option...

Pick the CORRECT eigen value(s) of the matrix $[\mathrm{A}]$ from the following choices. $$ [A]=\left[\begin{array}{ll} 6 & 8 \\ 4 & 2 \end{array}\right] $$

The statements P and Q are related to matrices A and B , which are conformable for both addition and multiplication. P: $(A + B)^T = A^T + B^T$ Q: $(AB)^T = B^T A^T$ Which one of the following options is CORRECT?

Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The magnitude of the minimum angle of rotation is

What are the eigenvalues of the matrix $\begin{bmatrix} 2 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 2 \end{bmatrix}$ ?

Consider two matrices $A = \begin{bmatrix}2 & 1 & 4 \\ 1 & 0 & 3\end{bmatrix}$ and $B = \begin{bmatrix}-1 & 0 \\ 2 & 3 \\ 1 & 4 \end{bmatrix}$. The determinant of the matrix $AB$ is __________ (in integer).

For the matrix $\rm [A]=\begin{bmatrix}1&-1&0\\\ -1&2&-1\\\ 0&-1&1\end{bmatrix}$ which of the following statements is/are TRUE?

Cholesky decomposition is carried out on the following square matrix [𝐴]. $\rm [A]=\begin{bmatrix}8&-5\\\ -5&a_{22}\end{bmatrix}$ Let 𝑙 ij and 𝑎ij be the (i, j) th elements of matrices [𝐿] and [𝐴], respectively. If...

If M is an arbitrary real n × n matrix, then which of the following matrices will have non-negative eigenvalues?

For the matrix $[A]= \begin{bmatrix}1&2&3\\\ 3&2&1\\\ 3&1&2 \end{bmatrix} $ which of the following statements is/are TRUE?

Two vectors [2 1 0 3] 𝑇 and [1 0 1 2] 𝑇 belong to the null space of a 4 × 4 matrix of rank 2. Which one of the following vectors also belongs to the null space?

P and Q are two square matrices of the same order. Which of the following statements is/are correct?

Both the numerator and the denominator of $${3 \over 4}$$ are increased by a positive $${15 \over 17}$$ integer, x, and those of _____ are decreased by the same integer. This operation results in the same value for both...

The components of pure shear strain in a sheared are given in the matrix form: $$\varepsilon = \left[ {\matrix{ 1 & 1 \cr 1 & { - 1} \cr } } \right]$$ Here, Trace ($$\varepsilon $$) = 0. Given, P = Trace ($$\varepsilon$$...

The matrix M is defined as $$M = \left[ {\matrix{ 1 & 3 \cr 4 & 2 \cr } } \right]$$ and has eigenvalues 5 and $$-$$2. The matrix Q is formed as Q = M 3 $$-$$ 4M 2 $$-$$ 2M Which of the following is/are the eigenvalue(s)...

Let y be a non-zero vector of size 2022 $$\times$$ 1. Which of the following statements is/are TRUE?

Consider the following equations of straight lines : Line L1 : 2x $$-$$ 3y = 5 Line L2 : 3x + 2y = 8 Line L3 : 4x $$-$$ 6y = 5 Line L4 : 6x $$-$$ 9y = 6 Which one of the following is the correct statement?

The Cartesian coordinates of a point P in a right-handed coordinate system are (1, 1, 1). The transformed coordinates of P due to a 45$$^\circ$$ clockwise rotation of the coordinate system about the positive x-axis are

Consider the following simultaneous equations (with $${c_1}$$ and $${c_2}$$ being constants): $$$3{x_1} + 2{x_2} = {c_1}$$$ $$$4{x_1} + {x_2} = {c_2}$$$ The characteristic equation for these simultaneous equation is

If $$A = \left[ {\matrix{ 1 & 5 \cr 6 & 2 \cr } } \right]\,\,and\,\,B = \left[ {\matrix{ 3 & 7 \cr 8 & 4 \cr } } \right]A{B^T}$$ is equal to

Consider the matrix $$\left[ {\matrix{ 5 & { - 1} \cr 4 & 1 \cr } } \right].$$ Which one of the following statements is TRUE for the eigenvalues and eigenvectors of this matrix?

The matrix $$P$$ is the inverse of a matrix $$Q.$$ If $${\rm I}$$ denotes the identity matrix, which one of the following options is correct?

Consider the following linear system $$$x+2y-3z=a$$$ $$$2x+3y+3z=b$$$ $$$5x+9y-6z=c$$$ This system is consistent if $$a,b$$ and $$c$$ satisfy the equation

If the entries in each column of a square matrix $$M$$ add up to $$1$$, then an eigenvalue of $$M$$ is

Let $$A = \left[ {{a_{ij}}} \right],\,\,1 \le i,j \le n$$ with $$n \ge 3$$ and $${{a_{ij}} = i.j.}$$ The rank of $$A$$ is :

The smallest and largest Eigen values of the following matrix are : $$\left[ {\matrix{ 3 & { - 2} & 2 \cr 4 & { - 4} & 6 \cr 2 & { - 3} & 5 \cr } } \right]$$

For what value of $$'p'$$ the following set of equations will have no solutions? $$$2x+3y=5$$$ $$$3x+py=10$$$

The two Eigen Values of the matrix $$\left[ {\matrix{ 2 & 1 \cr 1 & p \cr } } \right]$$ have a ratio of $$3:1$$ for $$p=2.$$ What is another value of $$'p'$$ for which the Eigen values have the same ratio of $$3:1$$?

The rank of the matrix $$\left[ {\matrix{ 6 & 0 & 4 & 4 \cr { - 2} & {14} & 8 & {18} \cr {14} & { - 14} & 0 & { - 10} \cr } } \right]$$ is

The determinant of matrix $$\left[ {\matrix{ 0 & 1 & 2 & 3 \cr 1 & 0 & 3 & 0 \cr 2 & 3 & 0 & 1 \cr 3 & 0 & 1 & 2 \cr } } \right]$$ is

With reference to the conventional Cartesian $$(x,y)$$ coordinate system, the vertices of a triangle have the following coordinates: $$\,\left( {{x_1},{y_1}} \right) = \left( {1,0} \right);\,\,\,\left( {{x_2},{y_2}} \rig...

The sum of Eigen values of the matrix, $$\left[ M \right]$$ is where $$\left[ M \right] = \left[ {\matrix{ {215} & {650} & {795} \cr {655} & {150} & {835} \cr {485} & {355} & {550} \cr } } \right]$$

Given the matrices $$J = \left[ {\matrix{ 3 & 2 & 1 \cr 2 & 4 & 2 \cr 1 & 2 & 6 \cr } } \right]$$ and $$K = \left[ {\matrix{ 1 \cr 2 \cr { - 1} \cr } } \right],\,\,$$ the product $${K^T}JK$$ is ______.

What is the minimum number of multiplications involved in computing the matrix product $$PQR?$$ Matrix $$P$$ has $$4$$ rows and $$2$$ columns, matrix $$Q$$ has $$2$$ rows and $$4$$ columns and matrix $$R$$ has $$4$$ rows...

The eigen values of matrix $$\left[ {\matrix{ 9 & 5 \cr 5 & 8 \cr } } \right]$$ are

The inverse of the matrix $$\left[ {\matrix{ {3 + 2i} & i \cr { - i} & {3 - 2i} \cr } } \right]$$ is

A square matrix $$B$$ is symmetric if ____

The following system of equations $$$x+y+z=3,$$$ $$$x+2y+3z=4,$$$ $$$x+4y+kz=6$$$ will not have a unique solution for $$k$$ equal to

The eigenvalues of the matrix $$\left[ P \right] = \left[ {\matrix{ 4 & 5 \cr 2 & { - 5} \cr } } \right]$$ are

The product of matrices $${\left( {PQ} \right)^{ - 1}}P$$ is

For what values of $$\alpha $$ and $$\beta $$ the following simultaneous equations have an infinite number of solutions $$$x+y+z=5,$$$ $$$x+3y+3z=9,$$$ $$$x + 2y + \alpha z = \beta $$$

Given that one root of the equation $$\,{x^3} - 10{x^2} + 31x - 30 = 0\,\,$$ is $$5$$ then other roots are

The minimum and maximum eigen values of matrix $$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr } } \right]$$ are $$-2$$ and $$6$$ respectively. What is the other eigen value?

The inverse of $$2 \times 2$$ matrix $$\left[ {\matrix{ 1 & 2 \cr 5 & 7 \cr } } \right]$$ is

Solution for the system defined by the set of equations $$4y+3z=8, 2x-z=2$$ & $$3x+2y=5$$ is

For a given matrix $$A = \left[ {\matrix{ 2 & { - 2} & 3 \cr { - 2} & { - 1} & 6 \cr 1 & 2 & 0 \cr } } \right],$$ one of the eigen value is $$3.$$ The other two eigen values are

Consider the matrices $$\,{X_{4x3,}}\,\,{Y_{4x3}}$$ $$\,\,{P_{2x3}}.$$ The order of $$\,{\left[ {P{{\left( {{X^T}Y} \right)}^{ - 1}}{P^T}} \right]^T}$$ will be

Consider the following system of equations in three real variable $${x_1},$$ $${x_2}$$ and $${x_3}:$$ $$$2{x_1} - {x_2} + 3{x_3} = 1$$$ $$$3{x_1} + 2{x_2} + 5{x_3} = 2$$$ $$$ - {x_1} + 4{x_2} + {x_3} = 3$$$ This system o...

Consider a non-homogeneous system of linear equations represents mathematically an over determined system. Such a system will be

Consider the system of equations, $${A_{nxn}}\,\,{X_{nx1}}\,\, = \lambda \,{X_{nx1}}$$ where $$\lambda $$ is a scalar. Let $$\left( {{\lambda _i},\,\,{X_i}} \right)$$ be an eigen value and its corresponding eigen vector...

The eigen values of the matrix $$\left[ {\matrix{ 4 & { - 2} \cr { - 2} & 1 \cr } } \right]$$ are

Real matrices $$\,\,{\left[ A \right]_{3x1,}}$$ $$\,\,{\left[ B \right]_{3x3,}}$$ $$\,\,{\left[ C \right]_{3x5,}}$$ $$\,\,{\left[ D \right]_{5x3,}}$$ $$\,\,{\left[ E \right]_{5x5,}}$$ $$\,\,{\left[ F \right]_{5x1,}}$$ ar...

Given matrix $$\left[ A \right] = \left[ {\matrix{ 4 & 2 & 1 & 3 \cr 6 & 3 & 4 & 7 \cr 2 & 1 & 0 & 1 \cr } } \right],$$ the rank of the matrix is

Eigen values of the following matrix are $$\left[ {\matrix{ { - 1} & 4 \cr 4 & { - 1} \cr } } \right]$$

The eigen values of the matrix $$\left[ {\matrix{ 5 & 3 \cr 2 & 9 \cr } } \right]$$ are

The determinant of the following matrix $$\left[ {\matrix{ 5 & 3 & 2 \cr 1 & 2 & 6 \cr 3 & 5 & {10} \cr } } \right]$$

The product $$\left[ P \right]\,\,{\left[ Q \right]^T}$$ of the following two matrices $$\left[ P \right]\,$$ and $$\left[ Q \right]\,$$ where $$\left[ P \right]\,\, = \left[ {\matrix{ 2 & 3 \cr 4 & 5 \cr } } \right],\,\...

If $$A,B,C$$ are square matrices of the same order then $${\left( {ABC} \right)^{ - 1}}$$ is equal be

Consider the following two statements. $$(I)$$ The maximum number of linearly independent column vectors of a matrix $$A$$ is called the rank of $$A.$$ $$(II)$$ If $$A$$ is $$nxn$$ square matrix then it will be non-singu...

If $$A$$ is any $$nxn$$ matrix and $$k$$ is a scalar then $$\left| {kA} \right| = \alpha \left| A \right|$$ where $$\alpha $$ is

The equation $$\left[ {\matrix{ 2 & 1 & 1 \cr 1 & 1 & { - 1} \cr y & {{x^2}} & x \cr } } \right] = 0$$ represents a parabola passing through the points.

The number of terms in the expansion of general determinant of order $$n$$ is

Obtain the eigen values and eigen vectors of $$A = \left[ {\matrix{ 8 & -4 \cr 2 & { 2 } \cr } } \right].$$

The real symmetric matrix $$C$$ corresponding to the quadratic form $$Q = 4{x_1}{x_2} - 5{x_2}{x_2}$$ is

In matrix algebra $$AS=AT$$ ($$A,S,T,$$ are matrices of appropriate order) implies $$S=T$$ only if

If $$A$$ is a real square matrix then $$A{A^T}$$ is

If $$A$$ and $$B$$ are two matrices and $$AB$$ exists then $$BA$$ exists,

If the determinant of the matrix $$\left[ {\matrix{ 1 & 3 & 2 \cr 0 & 5 & { - 6} \cr 2 & 7 & 8 \cr } } \right]$$ is $$26,$$ then the determinant of the matrix $$\left[ {\matrix{ 2 & 7 & 8 \cr 0 & 5 & { - 6} \cr 1 & 3 & 2...

Inverse of matrix $$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 1 & 0 & 0 \cr } } \right]$$ is