ESE 2021 Technical Quiz || Linear algebra(I & II)

Consider the following statements:

1) The rank of a matrix is the order of largest non-vanishing minor of A.

2) For any two matrices A & B, ρ(AB) ≥ min {ρ(A), ρ(B)}.

3) For any two matrices A and B, ρ(A+B) ≤ ρ(A)+ρ(B).

Which of the following statements is/are correct?

Consider matrix , then determinant of Adj (Adj A) is _________.

The rank of matrix is ___________.

Consider the system of linear equation:

2x₂ + x₃ = – 2

3x₁ + 5x₂ – 5x₃ = 1

2x₁ + 4x₂ – 2x₃ = 2

The above system has _________.

If two eigen values of matrix  are 0 and 6, then the eigen vector corresponding to the largest eigen value of A is: