Overview
This lecture covers elementary matrices, their connection to row operations, invertibility criteria for square matrices, and an algorithm for finding a matrix inverse using row reduction.
Identity Matrix and Row Operations
- The identity matrix is a square matrix with ones on the diagonal and zeros elsewhere.
- Multiplying any matrix by the identity matrix returns the original matrix.
- Matrix multiplication is not commutative; the order of multiplication matters.
- Multiplying a matrix by another matrix with a changed row is equivalent to performing a row operation.
Elementary Matrices
- An elementary matrix is formed by performing a single row operation on the identity matrix.
- Row swapping, scaling a row, and adding multiples of one row to another correspond to specific elementary matrices.
- Each elementary matrix has an inverse that undoes the corresponding row operation.
- The inverse of a row swap is another swap; the inverse of scaling is dividing by the scaling factor; the inverse of a row addition is subtracting.
Invertibility of Square Matrices
- An n x n (square) matrix is invertible only if it can be row reduced to the identity matrix.
- "If and only if" statements require proving both directions: invertibility implies row-equivalence to identity, and vice versa.
- If a matrix is invertible, its reduced row echelon form is the identity matrix.
- If a matrix can be reduced to the identity using elementary row operations, it is invertible.
Algorithm for Finding an Inverse
- To find the inverse, augment the matrix with the identity matrix and perform row operations to turn the left side into the identity.
- Apply the same row operations to the right side; the right side then becomes the inverse matrix.
- Verify the result by multiplying the inverse by the original matrix to ensure it equals the identity matrix.
Key Terms & Definitions
- Identity Matrix — A square matrix with ones on the diagonal and zeros elsewhere.
- Elementary Matrix — A matrix obtained by applying one row operation to the identity matrix.
- Invertible Matrix — A square matrix that has a unique inverse, such that multiplying by its inverse yields the identity.
- Row-equivalent — Two matrices are row-equivalent if one can be transformed into the other using row operations.
Action Items / Next Steps
- Practice finding inverses using the row reduction algorithm.
- Show the dividing line between the matrix and identity when augmenting for inverse calculations.
- Review related textbook examples on elementary matrices and invertibility.