Test: Eigenvalues & Eigenvectors - 2
25 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Eigenvalues & Eigenvectors - 2
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For the matrix 
the eigen value corresponding to the eigenvector 
the eigen value corresponding to the eigenvector 
the eigen value are
For which value of x will the matrix given below become singular?
Let the given matrix be A. A is singular.
If a square matrix A is real and symmetric, then the eigenvaluesn
The matrix 
has one eigenvalue equal to 3. The sum of the other two eigenvalues is
Let the given matrix be A.
For a matrix 
the transpose of the matrix is equal to the inverse of the matrix, 
The value of x is given by
For a given matrix 
one of the eigenvalues is 3. The other two eigenvalues are
In the matrix equation Px = q which of the following is a necessary condition for the existence of at least one solution for the unknown vector x:
then top row of R-1 is 
Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix
A satisfies the relation
Characteristic equation of A is
The characteristic equation of a (3×3) matrix P is defined as 
If I denote identity matrix, then the inverse of matrix P will be
Given ch. equn of A is
Let P be a 2×2 real orthogonal matrix and 
s a real vector 
with length 
Then which one of the following statements is correct?
Eigen values of P are 1,2,3
Let A be an n × n real matrix such that A2 = I and y = be an n – dimensional vector. Then the linear system of equations Ax = y has
By Cramer’s rule AX =y has unique solution.
A real n × n matrix A = {aij} is defined as follows:
aij = i = 0, if
i = j, otherwise
The summation of all n eigen values of A is
It’s a diagonal marix diagonal contain’s n elements 1,2,----,n.
As diagonal elements are eigen valves.
The following system of equations
has a unique solution. The only possible value(s) for a is/are
System has unique Soln if rank (A) = rank ( A ) = 3 . It is possible if a ≠ 5.
The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix. Hence 5, -19, and 37 are the eigenvalues of the matrix. Alternately, look at 
Then 
= 5,-19,37 are the roots of the equation; and hence, the eigenvalues of [A].
The number of different n × n symmetric matrices with each element being either 0 or 1 is: (Note : power (2, x) is same as 2x)
In a symmetric matrix, the lower triangle must be the minor image of upper triangle using the diagonal as mirror. Diagonal elements may be anything. Therefore, when we are counting symmetric matrices we count how many ways are there to fill the upper triangle and diagonal elements. Since the first row has n elements, second (n – 1) elements, third row (n – 2) elements and so on upto last row, one element. Total number of elements in diagonal + upper triangle
Now, each one of these elements can be either 0 or 1. So that number of ways we can fill these elements is
Since there is no choice for lower triangle elements the answer is power 
which
is choice (c).
In an M × N matrix such that all non-zero entries are covered in a rows and b column. Then the maximum number of non-zero entries, such that no two are on the same row or column, is
Suppose a < b, for example let a = 3, b= 5, then we can put non-zero entries only in 3 rows and 5 columns. So suppose we put non-zero entries in any 3 rows in 3 different columns. Now we can’t put any other non-zero entry anywhere in matrix, because if we put it in some other row, then we will have 4 rows containing non-zeros, if we put it in one of those 3 rows, then we will have more than one non-zero entry in one row, which is not allowed.
So we can fill only “a” non-zero entries if a < b, similarly if b < a, we can put only “b” non-zero entries. So answer is ≤min(a,b), because whatever is less between a and b, we can put atmost that many non-zero entries.
Consider the following system of equation in three real variables x1, x2 and x3
This system of equations has
∴ Rank (A)= Rank ( A ) = 3
How many of the following matrics have an eigenvalue 1?
Rest given matrix are triangular matrix. so diagonal elements are the eigen values.
What are the eigen values of the following 2 × 2 matrix?
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27 docs|243 tests
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27 docs|243 tests
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