# GATE Questions & Answers of Linear Algebra Mechanical Engineering

#### Linear Algebra 24 Question(s)

The rank of the matrix $\begin{bmatrix}-4&\;\;\;\;1&-1\\-1&-1&-1\\\;\;\;7&-3&\;\;\;1\;\end{bmatrix}$ is

If A = $\begin{bmatrix}1&2&3\\0&4&5\\0&0&1\end{bmatrix}$ then det$\left(A^{-1}\right)$ is __________ (correct to two decimal places).

The product of eigenvalues of matrix P is

$\mathrm P=\begin{bmatrix}2&0&1\\4&-3&3\\0&2&-1\end{bmatrix}$

Consider the matrix $\style{font-family:'Times New Roman'}{\mathrm P=\begin{bmatrix}\frac1{\sqrt2}&0&\frac1{\sqrt2}\\0&1&0\\\frac{-1}{\sqrt2}&0&\frac1{\sqrt2}\end{bmatrix}}$

Which one of the following statements about P is INCORRECT?

The determinant of a 2x2 matrix is 50. If one eigenvalue of the matrix is 10, the other eigenvalue is_________

Consider the matrix A=$\left[\begin{array}{cc}50& 70\\ 70& 80\end{array}\right]$ whose eigenvectors corresponding to eigenvalues ${\lambda }_{1}$ and ${\lambda }_{2}$ are ${\mathrm{X}}_{1}=\left[\begin{array}{cc}70& \\ {\lambda }_{1}& -50\end{array}\right]$ and ${\mathrm{X}}_{2}=\left[\begin{array}{cc}{\lambda }_{2}& -80\\ 70& \phantom{\rule{0ex}{0ex}}\end{array}\right]$ , respectively. The value of x1T x2 is___________

A swimmer can swim 10 km in 2 hours when swimming along the flow of a river. While swimming against the flow, she takes 5 hours for the same distance . Her speed in still water (in km/h) is______.

Given that the determinant of the matrix $\left[\begin{array}{ccc}1& 3& 0\\ 2& 6& 4\\ -1& 0& 2\end{array}\right]$ is -12, the determinant of the matrix $\left[\begin{array}{ccc}2& 6& 0\\ 4& 12& 8\\ -2& 0& 4\end{array}\right]$ is

The matrix form of the linear system $\frac{\mathrm{dx}}{\mathrm{dt}}=3x-5y$ and $\frac{\mathrm{dy}}{\mathrm{dt}}=4x+8y$ is

One of the eigenvectors of the matrix $\left[\begin{array}{cc}-5& 2\\ -9& 6\end{array}\right]$

Consider a 3×3 real symmetric matrix S such that two of its eigenvalues are $a\ne 0,b\ne 0$ with respective eigenvectors $\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\end{array}\right],$$\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\\ {y}_{3}\end{array}\right]$.If $a\ne b$ then ${x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}$ equals

Which one of the following equations is a correct identity for arbitrary 3×3 real matrices P, Q and R?

The eigenvalues of a symmetric matrix are all

For the matrix $A=\left[\begin{array}{cc}5& 3\\ 1& 3\end{array}\right]$, ONE of the normalized eigen vectors is given as

x + 2y + z = 4

2x + y + 2z = 5

x - y + z = 1

The system of algebraic equations given above has

Eigenvalues of a real symmetric matrix are always

Consider the following system of equations:
2x1 + x2 + x3 = 0,
x2 - x3 = 0,
x1 + x2 = 0.
This system has

One of the eigen vectors of the matrix $A=\left[\begin{array}{cc}2& 2\\ 1& 3\end{array}\right]$ is

For a matrix [M]= $\left[\begin{array}{cc}\frac{3}{5}& \frac{4}{5}\\ x& \frac{3}{5}\end{array}\right]$, the transpose of the matrix is equal to the inverse of the matrix [M]T = [M]-1. The value of x is given by

The matrix  $\left[\begin{array}{ccc}1& 2& 4\\ 3& 0& 6\\ 1& 1& p\end{array}\right]$ has one eigen value equal to 3. The sum of the other two eigenvalues is

For what value of a, if any, will the following system of equations in x,y and z have a solution?

2x + 3y = 4
x + y+ z = 4
x + 2y-z =a

The eigenvectors of the matrix $\left[\begin{array}{cc}1& 2\\ 0& 2\end{array}\right]$ are written in the form$\left[\begin{array}{c}1\\ a\end{array}\right]$ and $\left[\begin{array}{c}1\\ b\end{array}\right]$ What is a + b?

The number of linearly independent eigenvectors of $\left[\begin{array}{cc}2& 1\\ 0& 2\end{array}\right]$ is