Test: Engineering Mathematics- 3
10 Questions MCQ Test GATE Computer Science Engineering(CSE) 2023 Mock Test Series | Test: Engineering Mathematics- 3
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The matrix 
has one eigenvalue equal to 3. The sum of the other two eigenvalues is
has one eigenvalue equal to 3. The sum of the other two eigenvalues isSum of the eigen values of matrix is = Sum of diagonal values present in the matrix
∴ 1 + 0 + P = 3 + λ2 + λ3
⇒ P + 1 = 3 + λ2 + λ3
⇒ λ2 + λ3 = P + 1 – 3 = P – 2
What is the determinant of matrix X if 4 and (2 + 7i) are the eigenvalues of X where i = √−1?
Two eigen value of X is 4 and (2 + 7i)
∴ (2 - 7i) (conjugate roots) must be the third root
Determinant of P = product of eigenvalues
Δ = 4 × (2 + 7i) × (2 - 7i)
Δ = 212
In the given matrix
one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are
one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 areFor a given matrix A if V is the eigen vector corresponding to the eigen value λ, then:
AV = λV
∴ {α (−4, 2, 1) |α ≠ 0, αϵR} are the corresponding eigenvectors.
Which of the below-given statements is/are true?
I. The eigenvalue of the lower triangular matrix is just the diagonal elements of the matrix.
II. The product of the eigenvalue of a matrix is equal to its trace.
III. If 1/λ is an eigenvalue of A’(inverse of A) then orthogonal of A also have 1/λ as its eigenvalue.
- The eigenvalue of the triangular (lower or upper) matrix are just the diagonal elements of matrix.
- The product of the eigenvalue of a matrix is equal to its determinants.
- If λ is the eigenvalue of matrix, then 1/λ is the eigenvalue of its inverse since orthogonal is equal to inverse matrix then it has 1/λ as its eigenvalue
Example:
Eigenvalues are 1, 4 and 6 (diagonal elements)
Product of eigen value = determinants = 1 × 4 × 6 = 24
Orthogonal matrix and Inverse of given matrix have eigenvalues: 1,1/4 and 1/6
Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigenvalues of this matrix are - 1 and 7. What are the values of a and b?
W.K.T.∑λi = ∑aii
∴ - 1 + 7 = 1 + a ⇒ a = 5
Also π λi = |A|
∴ - 1 × 7 = a – 4b
- 7 = 5 – 4b ⇒ b = 3
Let A be the 2 X 2 matrix with elements a11 = 2, a12 = 3, a21 = 1 and a22 = 4 then the eigenvalues of the Matrix are A5?
Given Matrix:
Characteristic Equation:
(2−λ) (4−λ) −3 = 0
λ2 − 6λ + 5 = 0
λ = 5 or λ = 1
The latent values of the matrix
are 1, 1, 2 and the number of linearly independent latent vectors for the repeated root 1 is –
Eigen vectors are also called invariant vectors, characteristic vectors, or latent vectors. Eigenvalues are also called characteristic roots or latent roots.
(A - λI)X = 0
Given λ = 1
⇒ (A - I) X = 0
⇒ x = 2y + 3x = 0, 10x – 5y + 5z = 0, 5x – 4y + 5z = 0
By solving above equations, we get x =
i.e. one linearly independent latent vector.
Consider a Matrix M = uTvT where u = (112) and v = 
also uT denotes the transpose of matrix u. Find the largest eigenvalue of M?
Characteristic equation is given
λ3 − 5λ2 = 0
λ = 0 or λ = 5
Therefore largest value is 5.
What is the absolute difference of the eigenvalues for the matrix 
ad – bc = 6 and a + d = 7?
Let λ1 and λ2 be the two eigen values
Product of eigen value is equal to determinant of matrix
Sum of eigen value is equal to its trace
a + d = λ1 + λ2 = 7
λ1 × (7 − λ1) = 6
λ12 − 7λ + 6 = 0
λ1 = 6 or λ2 = 1
|λ1 − λ2| = 5
Consider the matrix 
which one of the following statements is TRUE for the eigenvalues and eigenvectors of the matrix?
(5 - λ) (1 - λ) + 4 = 0
5 – 5λ - λ + λ2 + 4 = 0
λ2 - 6λ + 9 = 0
(λ - 3)2 = 0
λ = 3
for eigen vector
So, only one independent eigen vector.
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