Signals & Systems Multiple Choice Questions on “Eigenvalues”.
1. Find the Eigen values of matrix (A = begin{bmatrix}
2 & 1 & 0\
1 & 2 & 1\
0 & 1 & 2\
end{bmatrix}).
A. 2 + (sqrt{2}), 2-(sqrt{2}), 2
B. 2, 1, 2
C. 2, 2, 0
D. 2, 2, 2
Answer: A
Clarification: To find the Eigen values it satisfy the condition, |A-λI|=0
|A-λI| = (begin{bmatrix}
2 & 1 & 0\
1 & 2 & 1\
0 & 1 & 2\
end{bmatrix} – λbegin{bmatrix}
1 & 0 & 0\
0 & 1 & 0\
0 & 0 & 1\
end{bmatrix})
|A-λI| = (begin{vmatrix}
2-λ & 1 & 0\
1 & 2-λ & 1\
0 & 1 & 2-λ\
end{vmatrix})
= 2 – (λ2-4λ+3) – (2-λ)
By solving the above equation, we get,
λ = 2 + (sqrt{2}), 2-(sqrt{2}), 2.
2. Find the product of Eigen values of a matrix (A = begin{bmatrix}
1 & 2 & 4\
0 & 6 & 0\
3 & 1 & 2\
end{bmatrix}).
A. 60
B. 45
C. -60
D. 40
Answer: C
Clarification: According to the property of Eigen values, the product of the Eigen values of a given matrix is equal to the determinant of the matrix |A| = 1(12-0) – 2(0) + 4(8)
= -60.
3. Let us consider a square matrix A of order n with Eigen values of a, b, c then the Eigen values of the matrix AT could be.
A. a, b, c
B. -a, -b, -c
C. a-b, b-a, c-a
D. a-1, b-1, c-1
Answer: A
Clarification: According to the property of the Eigen values, any square matrix A and its transpose AT have the same Eigen values.
4. What is Eigen value?
A. A vector obtained from the coordinates
B. A matrix determined from the algebraic equations
C. A scalar associated with a given linear transformation
D. It is the inverse of the transform
Answer: C
Clarification: Eigen values is a scalar associated with a given linear transformation of a vector space and having the property that there is some nonzero vector which is when multiplied by the scalar is equal to the vector obtained by letting the transformation operate on the vector.
5. Find the sum of the Eigen values of the matrix (A = begin{bmatrix}
3 & 6 & 7\
5 & 4 & 2\
7 & 9 & 1\
end{bmatrix}).
A. 7
B. 8
C. 9
D. 10
Answer: B
Clarification: According to the property of the Eigen values, the sum of the Eigen values of a matrix is its trace that is the sum of the elements of the principal diagonal.
Therefore, the sum of the Eigen values = 3 + 4 + 1 = 8.
6. Let the matrix A be the idempotent matrix then the Eigen values of the idempotent matrix are ________
A. 0, 1
B. 0
C. 1
D. -1
Answer: A
Clarification: According to the property of the Eigen values, the Eigen values of the idempotent matrix are either zero or unity.
So, the answer is 0 or 1.
7. Let us consider a 3×3 matrix A with Eigen values of λ1, λ2, λ3 and the Eigen values of A-1 are?
A. λ1, λ2, λ3
B. ( frac{1}{λ_1}, frac{1}{λ_2}, frac{1}{λ_3})
C. -λ1, -λ2, -λ3
D. λ1, 0, 0
Answer: B
Clarification: According to the property of the Eigen values, if is the Eigen value of A, then ( frac{1}{λ}) is the Eigen value of A-1.
So the Eigen values of A-1 are ( frac{1}{λ_1}, frac{1}{λ_2}, frac{1}{λ_3}).
8. The Eigen values of a 3×3 matrix are λ1, λ2, λ3 then the Eigen values of a matrix A3 are __________
A. λ1, λ2, λ3
B. ( frac{1}{λ_1}, frac{1}{λ_2}, frac{1}{λ_3})
C. (λ_1^3, λ_2^3, λ_3^3)
D. 1, 1, 1