High School Mathematics Extensions/Matrices/Project/Elementary Matrices
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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 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 C^{T} where the ith row of C is the ith 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?