1 Crore+ students have signed up on EduRev. Have you? 
The matrix has one eigenvalue equal to 3. The sum of the other two eigenvalues is
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
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
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.
Which of the belowgiven 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.
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.∑λ_{i }= ∑a_{ii}
∴  1 + 7 = 1 + a ⇒ a = 5
Also π λ_{i} = A
∴  1 × 7 = a – 4b
 7 = 5 – 4b ⇒ b = 3
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}?
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 = u^{T}v^{T} where u = (112) and v = also u^{T} denotes the transpose of matrix u. Find the largest eigenvalue of M?
Characteristic equation is given
λ^{3} − 5λ^{2} = 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
λ1^{2} − 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 + λ^{2} + 4 = 0
λ^{2}  6λ + 9 = 0
(λ  3)^{2} = 0
λ = 3
for eigen vector
So, only one independent eigen vector.
3 videos7 docs100 tests

Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 
3 videos7 docs100 tests









