High School Mathematics Extensions/Matrices/Project/Elementary Matrices

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Project -- Elementary matricesEdit

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

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 CT where the ith row of C is the ith column of CT. 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?