Test: Eigenvalues & Eigenvectors - 1
20 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Eigenvalues & Eigenvectors - 1
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Find the sum of the Eigenvalues of the matrix
According to the property of the Eigenvalues, the sum of the Eigenvalues of a matrix is its trace that is the sum of the elements of the principal diagonal.
Therefore, the sum of the Eigenvalues = 3 + 4 + 1 = 8.
A - λI = 0
Now, After taking the determinant:
(4 - λ)2 - 1 = 0
16 + λ2 - 8λ - 1 = 0
λ2 - 8λ + 15 = 0
(λ - 3) (λ - 5) = 0
λ = 3, 5
All the four entries of the 2 × 2 matrix 
are nonzero, and one of its eigen values is zero. Which of the following statements is true?
are nonzero, and one of its eigen values is zero. Which of the following statements is true?One eigen value is zero
Let the matrix be A. We know, Trace (A) = sum of eigen values.
The three characteristic roots of the following matrix A
The sum of the eigenvalues of the matrix given below is
For which value of x will the matrix given below become singular?
Let the given matrix be A. A is singular.
Eigen values of a matrix 
are 5 and 1. What are the eigen values of the matrix S2 = SS?
We know If λ be the eigen value of A ⇒λ2 is an eigen value of A2 .
For S matrix, if eigen values are λ1, λ2, λ3,... then for S² matrix, the eigen values will be λ²1 λ²2 λ²3......
For S matrix, if eigen values are 1 and 5 then for S² matrix, the eigen values are 1 and 25
The number of linearly independent eigenvectors of 
Number of linear independent vectors is equal to the sum of Geometric Multiplicity of eigen values. Here only eigen value is 2.
To find Geometric multiplicity find n-r of (matrix-2I), where n is order and r is rank.
Rank of obtained matrix is 1 and n = 2 so n-r = 1. Therefore the no of linearly independent eigen vectors is 1
The eigenvectors of the matrix 
are written in the form 
. What is a + b?
One of the Eigenvectors of the matrix A = 
is
The eigen vectors of A are given by AX= λ X
So we can check by multiplication.
The minimum and the maximum eigen values of the matrix 
are –2 and 6, respectively. What is the other eigen value?
The state variable description of a linear autonomous system is, X= AX,
Where X is the two dimensional state vector and A is the system matrix given by 
The roots of the characteristic equation are
Characteristic equation will be :λ2 -4 =0 thus root of characteristic equation will be +2 and - 2.
For the matrix 
s one of the eigen values is equal to -2. Which of the following is an eigen vector?
x=[x1x2…..xn]T is an n-tuple nonzero vector. The n×n matrix V=xxT
Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix
A9 equals
If the rank of a (5×6) matrix Q is 4, then which one of the following statements is correct?
Rank of a matrix is equal to the No. of linearly independent row or no. of linearly independent column vector.
The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues are
Identify which one of the following is an eigenvector of the matrix 
Eigen Value (λ ) are 1,− 2.
be the eigen of A. Corresponding
To λ then.
be the eigen vector corrosponding to λ = 1
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 atd
λ = 5, -19, 37
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