Let be any ring and let be an integer. Prove that if is any strictly upper triangular matrix in then . (A strictly upper triangular matrix is a matrix whose entries on and below the main diagonal are zero.)

Before approaching this problem, we will introduce some “structural” operations on matrices and prove some basic properties.

Definition: Let be a set.

- Suppose and . We define a matrix as follows: if and otherwise.
- Suppose and . We define a matrix as follows: if and otherwise.

Lemma: Let be a set, , , , and . Then

Proof: Let denote the matrix on the left hand side of the equals sign and the matrix on the right. We consider four possibilities for .

- Suppose and . Then .
- Suppose and . Then .
- Suppose and . Then .
- Suppose and . Then .

Thus .

Since these two operators “abide”, we will drop the inner brackets and write (for example) for brevity.

Lemma: Let be a ring.

- If , , and , then .
- If , , and , then .

Proof: The entry of is . If , then this sum is . If , then this sum is . Thus for all . The proof of the second statement is analogous.

Lemma: Let be a ring. If , , , and , then . Proof: For each , note the following.

= | ||

= | ||

= | ||

= | ||

= |

Thus the two matrices are equal.

Lemma: Let be a ring. Let , , , , , , , and . Then

Proof: Using the previous lemmas, we have the following.

= | ||

= | ||

= | ||

= | ||

= | . |

We now introduce another definition.

Definition: Let be a ring, , and . A matrix is called -strictly upper triangular if where is the zero matrix, has dimensions , and is upper triangular.

For example, 1-strictly upper triangular matrices and strictly upper triangular matrices are the same, and an matrix is zero if and only if it is -strictly upper triangular.

Lemma: Let be a ring, , and a square matrix over of dimension . If is strictly upper triangular and , where is square, then is strictly upper triangular. Proof: The elements on or below the main diagonal of are on or below the main diagonal of , hence are zero.

Lemma: Let be a ring, , and . If is -strictly upper triangular and is strictly upper triangular, then is -strictly upper triangular. Proof: We have , where is upper triangular and has dimension . Thus we have the following.

So that is -strictly upper triangular.

Lemma: Let be a ring, let , and let . If is upper triangular and is strictly upper triangular, then is strictly upper triangular. Proof: Recall that . Suppose . Then if , . If , . Thus , so that is strictly upper triangular.

Lemma: Let be a ring, let , and let . If such that is -strictly upper triangular and is strictly upper triangular, then is -strictly upper triangular. Proof: Write and , where and have dimension . Evidently, . By the previous lemma, since is upper triangular and is strictly upper triangular, is strictly upper triangular. Thus is -strictly upper triangular.

Now to the main result.

If is an matrix over a ring , and is strictly upper triangular, then by an easy induction argument is -strictly upper triangular. Thus .

## Comments

It will be easier if we use the Cayley-Hamilton theorem.

True, but that hasn’t been proved yet. The ease of proving this result with C-H is a strong argument that this exercise is in the wrong place. đŸ˜›