Elementary Transformations

• Interchange of any columns or rows
• Addition of multiple of any row or column to any other row or column
• Multiplication of each element of a column or a row by a non-zero constant

When a matrix $B$ is obtained by applying elementary transformations to a matrix $A$, then $A$ is equivalent to $B$. Denoted by $A\approx B$.

Theorem

The elementary row operations that reduce a given matrix $A$ to the identity matrix, also transform the identity matrix to the inverse of $A$.

Augmented Matrix

Two matrices are written as a single matrix with a vertical line in-between. Denoted by $(A\lvert B)$. Example: $$

$$\left[ \begin{array}{cc|c} 1&2&3\\ 4&5&6 \end{array} \right]$$

Finding Inverse

Let $A$ be a square matrix with order $n\times n$.

• Start with $(A_{n\times n}|I_n)$
• Repeatedly perform row transformations (not column) to both matrices until the $\text{LHS}$ becomes an identity matrix.
  • Transform all elements outside the main diagonal to $0$.
  • Transform elements on the main diagonal to $1$ by multiplying by a constant.
• $\text{RHS}$ is $A^{-1}$.

For singular matrices

They don’t have an inverse, so the process of using elementary row transformations to find $A^{-1}$ will fail in such cases. $$

Typically occurs because at least one row becomes all zeros during the reduction process, indicating that the matrix has no full rank ($\text{rank}(A) < n$). $$