Square matrices

The (main) diagonal of a square matrix is formed by the elements a₁₁, a₂₂, …,a_nn. The trace is the addition of these elements:

Example:

The identity or unit matrix is:

and it verifies that it is the identity element of multiplication, that is:

A square matrix is said to be symmetric if  A^t  = A, and antisymmetric if A^t  = -A

Examples:

 

Exercise: let

a) Calculate their traces

b) Find x, y and z to make A symmetric

c) Find a, b and c to make B antisymmetric

 

 

 

 

Solutions: a) tr(A) = y + 16, tr(B) = c; b) x = -1, y € R, z = 0; c) a = -3, b = -1, c = 0