Linear Transform MCQ - 4 - Mathematics MCQ

The eigenvalues of the matrix 

For the matrix  one of the eigenvalues is 3. The other two eigenvalues are

Write Matrix corresponding to the following linear transformations.
y₁ = 2 x 1 - x₂ - x₃ 
y₂ = 3 x 3 
y₃ = x₁ + x₂

The eigenvalues of a skew symmetric matrix are

The minimal polynomial m(x) of A_nxn each of whose element is 1 is

The characteristic equation of a 3 x 3 matrix A is defined as C(λ) = |λ - Al| = λ³ + λ² + 2λ + 1 = 0. If l denotes identity matrix then the inverse of matrix A will be

Let A be area 4 x 4 matrix with characteristic polynomial C(x) = (x² + 1)² which of the following is true?

If A is 3 x 3 matrix over α, β, α ≠ β are the only characteristic roots (eigenvalues) of A in the characteristic polynomail of A is

If A is symmetric matrix λ₁,λ₂,.... ,λ_n be the eigenvalues of A and a₁₁,a₂₂,.....,a_nn is the diagonal entries of A. Then which of the following is correct?

Let (-, -) be a symmetric bilinear form on ℝ² such that there exist nonzero v, w ∈ ℝ2 such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true? 

A square matrix A is said to be idempotent if A² = A. An idempotent matrix is non singular iff

Let V and V' be vector spaces over a field F. Then for any t₁, t₂ ∈ Hom (V, V') 

 then the eigenvalues of A are

If A and B are 3 × 3 real matrices such that rank (AB) = 1, then rank (BA) cannot be 

Let T be a linear operator on a finite dimensional vector space V, then which of the following is false

Let A be a 2 x 2 real matrix of rank 1. If A is not diagonalizable then

are given vectors and A  and if P = [x₁   x₂] then P^-1AP