Eigenvalues & Eigenvectors - 1 - GATE MA Engineering Mathematics Free MCQ

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

One eigen value is zero

Let the matrix be A.  We know, Trace (A) = sum of eigen values. 

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 eigen vectors of A are given by  AX= λ X  

So we can check by multiplication.

Concept:

If A is any square matrix of order n, we can form the matrix [A – λI], where I is the nth order unit matrix.

The determinant of this matrix equated to zero

i.e. |A – λI| = 0 is called the characteristic equation of A.

The roots of the characteristic equation are called Eigenvalues or latent roots or characteristic roots of matrix A.

Characteristic equation will be :λ² -4 =0 thus root of characteristic equation will be +2 and - 2.

Answer: b) of modulus one

Explanation:

Since A is self-adjoint, its eigenvalues (denoted λ) are real.

Let v be an eigenvector with A v = λ v. Then
(A minus i I) times the inverse of (A plus i I), acting on v, gives
((λ minus i) divided by (λ plus i)) times v.

Therefore, every eigenvalue of (A − i I)(A + i I)⁻¹ has modulus one.

• Observe that the absolute value of (λ minus i) is √(λ² + 1), which is the same as the absolute value of (λ plus i). Hence the absolute value of (λ minus i)/(λ plus i) is 1.

Rank of a matrix is equal to the No. of linearly independent row or no. of linearly  independent column vector. 

Vector R may be written as

R = 32i + 16j + 64k = 16 (2i + j + 4k) = 16X

So vector X after transformation remains on its span and only its magnitude is enlarged by 16. Hence, one of eigenvalue of matrix A is 16.

Hence option D is the correct option.