Create a 3-by-3 matrix and calculate the sum of the diagonal elements.

A = [1 -5 2; 
    -3  7 9; 
     4 -1 6];

b = trace(A)

b = 14

The result $\mathrm{tr}\left(\mathit{A}\right)=14$ agrees with a manual calculation.

Verify several properties of the trace of a matrix (up to round-off error).

Create two matrices. Verify that $\mathrm{tr}\left(\mathit{A}+\mathit{B}\right)=\mathrm{tr}\left(\mathit{A}\right)+\mathrm{tr}\left(\mathit{B}\right)$.

A = magic(3);
B = rand(3);
trace(A+B) 

ans = 17.4046

trace(A) + trace(B)

ans = 17.4046

Verify that $\mathrm{tr}\left(\mathit{A}\right)=\mathrm{tr}\left({\mathit{A}}^{\mathit{T})}\right)$.

trace(A)

ans = 15

trace(A')

ans = 15

Verify that $\mathrm{tr}\left({\mathit{A}}^{\mathit{T}}\mathit{B}\right)=\mathrm{tr}\left({\mathrm{AB}}^{\mathit{T}}\right)$.

trace(A'*B) 

ans = 22.1103

trace(A*B')

ans = 22.1103

Verify that $\mathrm{tr}\left(\mathrm{cA}\right)=\mathit{c}\mathrm{tr}\left(\mathit{A}\right)$ for a scalar $\mathit{c}$.

c = 5;
trace(c*A) 

ans = 75

c*trace(A)

ans = 75

Verify that the trace equals the sum of the eigenvalues $\mathrm{tr}\left(\mathit{A}\right)={\sum }_{\mathit{i}}{\lambda }_{\mathit{i}}$.

trace(A)

ans = 15

sum(eig(A))

ans = 15.0000