## Induction With Matrices

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.