Mathematics - Vectors and Matrices

Replacing 

expanding with respect to R₁

= 2(-14) - (4)(-2)

= -28  + 8 ≠ 0  

 .............................................(i)

Also A does not possess any minor of order 4, i.e. 3 + 1

 ..................................................(ii)

From Equations (i) and (ii), we get

p(A) = 3 i.e. rank of A is 3.

λ=  1, 4, 4 are the eigen values.

This is a skew-symmetric matrix.

A^T=A

Let α₁ and α₂ be different eigen values of matrix A, and X₁ and X₂ be the corresponding vectors, then

AX₁= α₁X₁ and AX₂ = α₂X₂

Taking transpose of the second equation 

(AX₂)^T=  (α₂X₂)

X_2TA^T= α₂.X_{2T 2}X_2T

But A^T-A

Post multiply by X₁, we get

X^T₂AX₁ = a₂ X_2T X₁

But AX₁ = a₁X₁

  X^T₂ a₁X₁ = a₂ X_2T X₁

 (a₁ - a₂) X_2TX₁ = 0

Since a₁  a2, a₁ - a₂  0

 X_2TX₁ = 0 i.e. X₂ and X₁ are orthogonal.