Unlocking Matrix Inverses: How Elementary Matrices Simplify Linear Algebra

Elementary Matrices

Starting from the identity matrix, applying a single elementary transformation yields an elementary matrix. The three elementary transformations correspond to three types of elementary matrices:

Swap matrix: obtained by exchanging two rows (or columns) of the identity matrix.

Scaling matrix: obtained by multiplying a row (or column) of the identity matrix by a non-zero scalar.

Shear (addition) matrix: obtained by adding a multiple of one row to another row (or similarly for columns) of the identity matrix.

Theorem 1

Performing one elementary row operation on a matrix is equivalent to left-multiplying the matrix by the corresponding elementary matrix; performing one elementary column operation is equivalent to right-multiplying the matrix by the corresponding elementary matrix.

Note: “corresponding” means the elementary matrix that swaps the specific rows, scales a specific row, or adds a multiple of one row to another (or the analogous column operations).

Theorem 2

If A is an invertible matrix, then there exist a finite sequence of elementary matrices E₁, E₂, …, Eₖ such that Eₖ … E₂ E₁ A = I .

Corollary: A matrix A is invertible if and only if there exists a product of elementary matrices that equals A⁻¹ .

Example 1

Find the inverse of a given matrix using elementary transformations.

Solution: (details of the row operations leading to the inverse would be shown here).