Many properties of matrices of presereved by matrix multiplication.

If square diagonal matrices are multiplied (diagonal means that entries on the leading diagonal are non zero egthe result is a diagonal matrix.

If upper triangular matrices – example- are multiplied, the result is an upper triangular matrix and if lower triangular matrices are multiplied, the result is a lower triangular matrix.

Matrix proofs using induction often deal with powers of matrices.

Ifthen

Ifthen entry in the upper right corner ofis 2 and the diagonal entries are 1.

Ifthen entry in the upper right corner ofis 4 and the diagonal entries are 1.

Ifthen entry in the upper right corner ofis 5 and the diagonal entries are 1.

We might speculate that the entry in the upper right corner ofisand and the diagonal entries are 1 and we can prove this by induction. Supposeis the statement ' the entry in the upper right corner ofis'.

Ifthe upper right entry is 2 and the diagonal entries are 1 so the basis step is true.

Suppose P(n) is true so that

is true so the staement is proved by induction.