Questions & Answers of Linear Algebra

Let c1,....,cn be scalars, not all zero, such that $\style{font-family:'Times New Roman'}{\sum_\limits{i=1}^nc_ia_i=0}$ where ai are column vectors in Rn Consider the set of linear equations
                                                      $\style{font-family:'Times New Roman'}{Ax=b}$
where $\style{font-family:'Times New Roman'}{A=\left[a_1,....,a_n\right]\;\mathrm{and}\;b=\sum_\limits{i=1}^na_i}$. The set of equations has 

Let u and v be two vectors in R2 whose Euclidean norms satisfy $\style{font-family:'Times New Roman'}{\parallel u\parallel=2\parallel v\parallel.}$ What is the value of  $\style{font-family:'Times New Roman'}\alpha$ such that  $\style{font-family:'Times New Roman'}{w=u+\alpha v}$ bisects the angle between u and v ?

Let A be $\style{font-family:'Times New Roman'}{n\times n}$ real valued  square symmetric matrix of rank 2 with $\style{font-family:'Times New Roman'}{\sum_\limits{i=1}^n\sum_\limits{j=1}^nA_{ij}^2=50.}$ Consider the following statements.
 
(I) One eigenvalue must be in [-5, 5]
 
(II) The eigenvalue with the largest magnitude must be strictly greater than 5
 
Which of the above statments about eigenvalue of A is/are necessarily CORRECT?
 

Let $ P=\begin{bmatrix}1&1&-1\\2&-3&4\\3&-2&3\end{bmatrix} $ and $ Q=\begin{bmatrix}-1&-2&-1\\6&12&6\\5&10&5\end{bmatrix} $ be two matrices.

Then the rank of P+Q is ________.

If the characteristics polynomial of a $\style{font-family:'Times New Roman'}{3\times3}$ matrix M over $\mathbb{R}$ (the set of real numbers) is $\style{font-family:'Times New Roman'}{\lambda^3-4\lambda^2+a\lambda+30,\;a\in\mathbb{R}}$ and one eigenvalue of M is 2, then the largest among the absolute values of the eigenvalues of M is _____________. 

Suppose that the eigenvalues of matrix A are 1,2,4. The determinant of $(A^{-1})^T$ is __________.

The larger of the two eigenvalues of the matrix 4521 is _____.

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?

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 __________________

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

If the matrix A is such that

A=2-47195

then the determinant of A is equal to ______.

The product of the non-zero eigenvalues of the matrix

1000101110011100111010001

is ______.

Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?

If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1V2 is ______.

Which one of the following does NOT equal 1xx21yy21zz2?

Let A be the 2 × 2 matrix with elements a11 = a12 = a21 = +1 and a22 = −1. Then the eigenvalues of the matrix A19 are

Consider the matrix as given below.

123047003

Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?

Consider the following matrix

A=23xy

If the eigenvalues of A are 4 and 8, then

The following system of equations

x1+x2+2 x3 = 1
 x1+2 x2+3 x3 = 2
x1+4 x2+
α x3 = 4

has a unique solution. The only possible value(s) for α is/are

How many of the following matrices have an eigenvalue 1?

1000,0100,1-111  and -101-1

Let A be a 4 × 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of AIIA, where I is the 4 × 4 identity matrix?

Consider the set of (column) vectors defined by X=xR3|x1+x2+x3=0, where xT=x1,x2,x3T. Which of the following is TRUE?

Subjects of Engineering Mathematics