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QUESTION: 1

The matrix has one eigenvalue equal to 3. The sum of the other two eigenvalues is

Solution:

Sum of the eigen values of matrix is = Sum of diagonal values present in the matrix

∴ 1 + 0 + P = 3 + λ_{2 }+ λ_{3}

⇒ P + 1 = 3 + λ_{2 }+ λ_{3}

⇒ λ_{2 }+ λ_{3 }= P + 1 – 3 = P – 2

QUESTION: 2

What is the determinant of matrix X if 4 and (2 + 7i) are the eigenvalues of X where i = √−1?

Solution:

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**

QUESTION: 3

In the given matrix one of the eigenvalues is 1. The eigenvectors corresponding to the eigenvalue 1 are

Solution:

For 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.

QUESTION: 4

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.

Solution:

- 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

QUESTION: 5

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?

Solution:

W.K.T.∑λ_{i }= ∑a_{ii}

∴ - 1 + 7 = 1 + a ⇒ a = 5

Also π λ_{i} = |A|

∴ - 1 × 7 = a – 4b

- 7 = 5 – 4b ⇒ b = 3

QUESTION: 6

Let A be the 2 X 2 matrix with elements a_{11} = 2, a_{12} = 3, a_{21} = 1 and a_{22} = 4 then the eigenvalues of the Matrix are A^{5}?

Solution:

Given Matrix:

Characteristic Equation:

(2−λ) (4−λ) −3 = 0

λ2 − 6λ + 5 = 0

λ = 5 or λ = 1

*Answer can only contain numeric values

QUESTION: 7

The latent values of the matrix

are 1, 1, 2 and the number of linearly independent latent vectors for the repeated root 1 is –

Solution:

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.

*Answer can only contain numeric values

QUESTION: 8

Consider a Matrix M = u^{T}v^{T} where u = (112) and v = also u^{T} denotes the transpose of matrix u. Find the largest eigenvalue of M?

Solution:

Characteristic equation is given

λ^{3} − 5λ^{2} = 0

λ = 0 or λ = 5

Therefore largest value is 5.

QUESTION: 9

What is the absolute difference of the eigenvalues for the matrix ad – bc = 6 and a + d = 7?

Solution:

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

λ1^{2} − 7λ + 6 = 0

λ1 = 6 or λ2 = 1

|λ1 − λ2| = 5

QUESTION: 10

Consider the matrix which one of the following statements is TRUE for the eigenvalues and eigenvectors of the matrix?

Solution:

(5 - λ) (1 - λ) + 4 = 0

5 – 5λ - λ + λ^{2} + 4 = 0

λ^{2} - 6λ + 9 = 0

(λ - 3)^{2} = 0

λ = 3

for eigen vector

So, only one independent eigen vector.

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