## 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 eg the 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.

If then  If then entry in the upper right corner of is 2 and the diagonal entries are 1.

If then entry in the upper right corner of is 4 and the diagonal entries are 1.

If then entry in the upper right corner of is 5 and the diagonal entries are 1.

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

If the 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. 