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When computing the inverse of a matrix, elementary row operations are fundamental. These operations are simple manipulations that we perform on the rows of a matrix. There are three types of elementary row operations:

• Switching two rows.
• Multiplying a row by a non-zero scalar.
• Adding a multiple of one row to another row.

These operations are crucial because they allow us to transform a matrix into different forms, such as the identity matrix, which is essential when trying to determine the inverse.
By performing these operations, we can methodically simplify the matrix and change its form while preserving its essential properties. This is how we verify if the original matrix is invertible.

The concept of an augmented matrix is instrumental when finding the inverse of a matrix using row operations. An augmented matrix combines two matrices into a single matrix representation. In the context of finding an inverse, we combine the original matrix with the identity matrix of the same size.
For example, if we have a 3x3 matrix, we augment it with a 3x3 identity matrix, resulting in a single matrix with 3 rows and 6 columns.

• The left half contains the original matrix.
• The right half contains the identity matrix.

This setup allows us to perform row operations simultaneously on both matrices. If we successfully transform the left matrix into the identity matrix, the right side will become the inverse if one exists. However, as shown in the solution, if we cannot achieve this transformation, it indicates that the matrix is singular.

An upper triangular matrix is a special form of a square matrix where all elements below the main diagonal are zero. When using row operations to find an inverse, one of the key steps is attempting to transform the matrix on the left side of the augmented matrix into an upper triangular form.
This transformation simplifies further operations, making it easier to assess the invertibility of the matrix.

• It involves creating zeros below each leading entry, or pivot, in each column.
• Once the upper triangular form is achieved, it helps in back substitution processes.

In our case, transforming the matrix didn't produce an upper triangular form beyond the first two rows. The entire third row turned to zero, revealing that the original matrix does not have the required full rank for an inverse, confirming it as singular.