Display each of the matrices
A, B, and C, together with its trace. Next enter the
diagonal matrix F, and display its trace as well. By comparing each
matrix to its trace, formulate a rule on how the trace may be calculated
in terms of the entries of the matrix.
Use a random square matrix
G to check your rule. Recompute the random matrix, and test your
rule several times.
Now look at the eigenvalues
of the matrices A, B, C, and F, and compare
them to the corresponding trace of each matrix. Formulate a rule relating
the eigenvalues of a matrix to its trace.
Check your rule with random
symmetric matrices of varying sizes.
Check your rule with random
matrices of varying sizes, not necessarily symmetric.
Explain why your rule must
be true for for the special case of diagonal matrices.