Inverses and Elementary Matrices
Part 2: Elementary Matrices and Row Operations
- Enter the definition in your worksheet for the 4 x 4 identity matrix. An elementary
matrix is any matrix that can be constructed from an identity matrix by
a single row operation. Enter the examples E1, E2, E3 defined in your worksheet.
Next, enter the "empty" symbolic matrix M. Compute each of the
products (E1)M, (E2)M, (E3)M, and describe the effect of left multiplication
by an elementary matrix.
- Find the inverse of each of the matrices E1,
E2, E3. Explain in general how to find the inverse of an elementary matrix
as another elementary matrix.
- Enter the new matrix A defined in your worksheet.
Reduce A to an upper triangular matrix U by left multiplications by elementary
matrices. (You can modify the definitions of E1, E2, E3. Name your elementary
matrices as you go so you can refer to them later.)
- Find a matrix P such that PA = U. In general,
for any two row equivalent matrices A and B, describe how to find a matrix
P such that PA = B. (Matrices A and B are row equivalent if there
is a sequence of elementary row operations that transforms A to B.)
- If Q is any invertible matrix, explain why Q
is row equivalent to an identity matrix. Then, with the help of the preceding
step, explain why Q is a product of elementary matrices.
- Combine results from the preceding steps to
prove the following theorem:
Two matrices A and B of the same size are row equivalent if and only
if there is an invertible matrix P such that PA = B.
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