Test: Linear Algebra - 4 - Mathematics MCQ

The determinant 

Which of Ihe following matrices has the same row space as the matrix 

The determinant of the n x n permutation 

[x] denotes greatest integer function of x.

The row space of a 20 x 50 matrix A has dimention 13. What is the dimension of the space of solution of Ax = 0?

Given a permutation the matrix A is defined to be the one whose i^th column is the σ(i)^th column of the Identity matrix I. Which of the following is correct?

For the matrix A as given below, which of them satisfy A⁶ = I?

Let I denote the 4 x 4 Identity matrix. If the roots of the characteristic polynomial of a 4 x 4 matrix M 

Let M be the set of all invertibel 5 x 5 matrices with entries 0 and 1. For each m ∈ M, let n₁ (m) and n₀(m) denote the number of 1's and 0's in m respectively then

 and b a non zero vector such that Mx = b for some Then the value of x^Tb is 

The matrix  is a unitary matrix when α is 

Let P be the vector space over all polynomials of degree less than 3 with real coefficients. Consider the linear transformation T : P → P defined by
T(a₀ + a₁x + a₂x² + a₃x³) = a₃ + a₂x + a₁x² + a₀x³

Then the matrix representation of M of T with respect to the ordered basis {1, x, x²,x³} satisfies 

The largest eigenvalue of the matrix 

Let a,b,c,d be distinct non zero real numbers with a + b = c + d. Then an eigenvalue of the matrix 

The system of linear equations 
x - y + 2z = b₁ 
x + 2y - z = b₂
2y - 2z = b₃
is inconsistent when (b₁, b₂, b₃) equals

Let X and Y are n x n matrices with real entries, then which of the following is(are) true?