Elementary Matrices and Matrix Inverses
Every elementary row operation corresponds to left-multiplication by a uniquely invertible elementary matrix.
Row Operations as Multiplication
An elementary matrix is created by applying a single elementary row operation to the identity matrix. Multiplying any matrix on the left by an elementary matrix executes that exact row operation on the target matrix. This allows us to express the process of row reduction as a sequence of matrix multiplications.
E A = A'Invertibility of Elementary Matrices
Because every elementary row operation can be undone by another elementary row operation of the same type, every elementary matrix is invertible. The inverse of an elementary matrix is simply the elementary matrix that represents the reversing operation. Consequently, products of elementary matrices are also always invertible.
E^{-1} E = IConstructing the Inverse Matrix
If a square matrix can be reduced to the identity matrix through a sequence of row operations, the matrix is invertible. Multiplying the identity matrix by that exact same sequence of elementary matrices produces the inverse matrix. This provides the formal proof for why Gauss-Jordan elimination successfully computes matrix inverses.
E_k \cdots E_2 E_1 A = I \implies A^{-1} = E_k \cdots E_2 E_1Flashcard
Matrix representing adding c times row j to row i
Check your understanding
What is the inverse of the 3 by 3 elementary matrix E that multiplies row 2 by 5?