High School Mathematics Extensions/Matrices/Project/Elementary Matrices

Project -- Elementary matrices 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?