Test: Eigenvalues & Eigenvectors - 2 - Mechanical Engineering MCQ

Let the given matrix be A.  A is singular. 

Characteristic equation of A is  

=> (λ + 1)(λ +2)
By Cayley theorem (A + 1I₂ ) ( A + 2I₂ )

Given ch. equⁿ of A is 

Eigen values of P are 1,2,3 

By Cramer’s rule AX =y has unique solution. 

 It’s a diagonal marix diagonal contain’s  n elements 1,2,----,n. 

As diagonal elements are eigen valves.

 System has unique Soln if  rank (A) = rank ( A ) = 3 . It is possible if a ≠ 5.

The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix. Hence 5, -19, and 37 are the eigenvalues of the matrix. Alternately, look at 

Then  = 5,-19,37 are the roots of the equation; and hence, the eigenvalues of [A].

In a symmetric matrix, the lower triangle must be the minor image of upper triangle using the diagonal as mirror. Diagonal elements may be anything. Therefore, when we are counting symmetric matrices we count how many ways are there to fill the upper triangle and diagonal elements. Since the first row has n elements, second (n – 1) elements, third row (n – 2) elements and so on upto last row, one element.  Total number of elements in diagonal + upper triangle  

Now, each one of these elements can be either 0 or 1. So that number of ways we can fill these elements is

Since there is no choice for lower triangle elements the answer is power   which 
is choice (c). 

Suppose a < b, for example let a = 3, b= 5, then we can put non-zero entries only in 3 rows and 5 columns. So suppose we put non-zero entries in any 3 rows in 3 different columns. Now we can’t put any other non-zero entry anywhere in matrix, because if we put it in some other row, then we will have 4 rows containing non-zeros, if we put it in one of those 3 rows, then we will have more than one non-zero entry in one row, which is not allowed.

So we can fill only “a” non-zero entries if a < b, similarly if b < a, we can put only “b” non-zero entries. So answer is ≤min(a,b), because whatever is less between a and b, we can put atmost that many non-zero entries.

∴ Rank (A)= Rank ( A ) = 3

Rest given matrix are triangular matrix. so diagonal elements are the eigen  values. 

Suppose λ₁ and λ₂ are eigenvalues of matrix A.

λ₁λ₂ = |A| = 2k - 1>0

λ₁ + λ₂ = 2+k >0

k>1/2 and k>-2

Both the conditions will be true for k>1/2

Hence option A is the correct answer.

For symmetric matrix having two distinct eigenvalues, corresponding eigenvectors are orthogonal.

x₁y₁ + x₂y₂ + x₃y₃ = 0

Hence option D is the correct answer.