Matrices

Let F: R³ →R² be the linear map defined by F(x, y, z) = (3x+2y-4z, x-5y+3z). The basis of R³ is S and basis of R² is S', where S = {(1, 1, 1), (1, 1, 0), (1, 0, 0)} and S' = {(1, 3), (2, 5)}. Then the matrix of F in the bases of R³ and R² is

If and then A+B =

The rank of matrix A =

Let A = then the matrix B that represents the linear operator A relative to the basis
S = { }= , is:

Which one of the following is a cyclic group?

The matrix P is the inverse of a matrix Q. If I denotes the identity matrix, which one of the following options is correct?

If the matrices form a group with respect to matrix multiplication, then which one of the following statements about the groups, thus formed is correct?

Consider the linear mapping F: R² →R² defined by F(x, y) = (3x+4y, 2x-5y) and following bases of R²: E= {e₁, e₂} = {(1, 0), (0, 1)} and S = {u₁, u₂} = {(1, 2), (2, 3)}. Then the matrix A representing F relative to the basis E is:

If λ₁,λ₂,λ₃ are the given values of the matrix , then λ₁²+λ₂²+λ₃² is equal to

The dimension of the general solution space W of the homogeneous system
x₁+2x₂-3x₃+2x₄-4x₅=0
2x₁+4x₂-5x₃+x₄-6x₅ = 0
5x₁+10x₂-13x₃+4x₄-16x₅= 0

If A= , then which one of the following is true?

Let A and B be 2 × 2 matrices, then which of the following is correct?

If A is symmetric real valued matrix of dimension 2022, then eigenvalues of A are