MATH 121A, Winter 2026 Discussion Sect. 3.1.

MATH 121A Discussion Section 3.1.

26 Feb 2026

1 Section 3.1. Elementary Matrices

1.1 Classwork

Exercise 1. What do the three types of elementary matrices do? Can you give an example of

3 × 3 elementary matrices for each type?

e.g.: Type 1: Interchanging two rows/columns. Example:  1 0 0

0 0 1

0 1 0



Solution.

ˆ Type 2 is multiplying a row/column by a nonzero scalar. Example:  1 0 0

0 2 0

0 0 1



.

ˆ Type 3 is adding any scalar multiple of a row/column to another. Example:  1 0 2

0 1 0

0 0 1



.

Exercise 2. What do each type of elementary matrices look like?

e.g.: Type 1: A symmetric matrix with 0 and 1 only and there are only two 1’s which are not

on the diagonal.

Solution.

ˆ Type 2: A diagonal matrix with non-zero diagonal entries, and at most one of the diagonal

entries is not 1.

ˆ Type 3: A square matrix with all diagonal entries equal to 1, and at most one non-diagonal

entry is non-zero.

Exercise 3 (If you have time). Determine if the following statements are true.

1. The sum of two elementary matrices is also elementary.

2. The non-zero scalar multiple of an elementary matrix is also elementary.

3. The product of two elementary matrices is also elementary.