# High School Mathematics Extensions/Matrices/Project/Elementary Matrices

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## Contents |

## Project -- Elementary matrices[edit]

Throughout,

1. The matrices below are called *elementary matrices*. How are the matrices below different from the identity matrix *I*, describe each one.

- where f is a scalar
- where f is a scalar
- where f is a scalar
- where f is a scalar

2. In each of the cases, compute **B** then describe how is **B** different from **A**

- where f is a scalar
- where f is a scalar
- where f is a scalar
- where f is a scalar

3. The matrix has determinant not equal to zero. We can *decompose* the matrix into products of elementary matrices pre-multiplying the identity:

Now suppose det(* A*) ≠ 0, can

*be expressed as the product of elementary matrices and the identity?*

**A**4. a) Show that every elementary matrix has an inverse. Hint: use determinant.

b) Prove that every invertible matrix (a matrix that has an inverse) is the product of some elementary matrices pre-multiplying the identity.

5. A *transpose* of a matrix * C* is the matrix

**C**^{T}where the

*i*th row of

*is the*

**C***i*th column of

**C**^{T}. Prove using elementary matrices that

for arbitrary matrices * D* and

*.*

**E**6. Show that every invertible matrix is also the product of some elementary matrices **post**-multiplying the identity.

7. How about non-invertible matrices? What can you say about them?