Linear Algebra · Engineering Mathematics · GATE CE

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...

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 TR...

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

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

The components of pure shear strain in a sheared are given in the matrix form: $$\varepsilon = \left[ {\matrix{ 1 & 1 \cr 1 & { - 1} \cr ...

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

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 form...

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$$^\circ$$ clock...

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

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

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

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 sum of Eigen values of the matrix, $$\left[ M \right]$$ is where $$\left[ M \right] = \left[ {\matrix{ {215} & {650} & {795} \cr ...

Given the matrices $$J = \left[ {\matrix{ 3 & 2 & 1 \cr 2 & 4 & 2 \cr 1 & 2 & 6 \cr } } \right]$$ and $$K = ...

The determinant of matrix $$\left[ {\matrix{ 0 & 1 & 2 & 3 \cr 1 & 0 & 3 & 0 \cr 2 & 3 & 0 & 1 \cr ...

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

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

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

In the solution of the following set of linear equations by Gauss-elimination using partial pivoting $$$5x+y+2z=34,$$$ $$$4y-3z=12$$$ and $$$10x-2y+z...

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

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

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

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{...

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

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

Consider the system of equations, $${A_{nxn}}\,\,{X_{nx1}}\,\, = \lambda \,{X_{nx1}}$$ where $$\lambda $$ is a scalar. Let $$\left( {{\lambda _i},\,\,...

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

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

Given matrix $$\left[ A \right] = \left[ {\matrix{ 4 & 2 & 1 & 3 \cr 6 & 3 & 4 & 7 \cr 2 & 1 & 0 & 1 ...

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

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

The product $$\left[ P \right]\,\,{\left[ Q \right]^T}$$ of the following two matrices $$\left[ P \right]\,$$ and $$\left[ Q \right]\,$$ where $$\left...

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

Consider the following two statements. $$(I)$$ The maximum number of linearly independent column vectors of a matrix $$A$$ is called the rank of $$A....

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

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

The equation $$\left[ {\matrix{ 2 & 1 & 1 \cr 1 & 1 & { - 1} \cr y & {{x^2}} & x \cr } } \right] = 0$$ repres...

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

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

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

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

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

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

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 de...

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

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 n...

Cholesky decomposition is carried out on the following square matrix [𝐴].  $\rm [A]=\begin{bmatrix}8&-5\\\ -5&a_{22}\end{bmatrix}$ Let...

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? ...

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

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

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 equ...

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

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 ...

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

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

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

The minimum and maximum eigen values of matrix $$\left[ {\matrix{ 1 & 1 & 3 \cr 1 & 5 & 1 \cr 3 & 1 & 1 \cr }...

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,$...

For a given matrix $$A = \left[ {\matrix{ 2 & { - 2} & 3 \cr { - 2} & { - 1} & 6 \cr 1 & 2 & 0 \cr } } \right...