91Ó°ÊÓ

An augmented matrix is a key concept when solving systems of linear equations using matrix operations. It basically combines the original matrix with another matrix. In problems involving finding the inverse of a matrix, the augmented matrix merges the given matrix with the identity matrix of the same dimension.
This combination helps in performing row operations easily and systematically. For example, in our exercise, we add the 4x4 identity matrix to the right of our given 4x4 matrix, represented as follows:
\[\begin{bmatrix} 1 & -2 & 3 & 0 & | & 1 & 0 & 0 & 0 \ 0 & 1 & -1 & 1 & | & 0 & 1 & 0 & 0 \ -2 & 2 & -2 & 4 & | & 0 & 0 & 1 & 0 \ 0 & 2 & -3 & 1 & | & 0 & 0 & 0 & 1 d\begin{bmatrix}\]
This is a crucial step for the elimination processes needed to find the inverse.

Gaussian elimination is a step-by-step method used to solve systems of linear equations. It reduces a matrix to its row-echelon form using three types of row operations:
• Swapping rows.
• Multiplying a row by a nonzero scalar.
• Adding or subtracting a multiple of one row from another row.
In our exercise, we use Gaussian elimination to simplify the augmented matrix. The goal here is to make the left side of the augmented matrix (the original matrix) into an identity matrix. This involves using row operations to make all elements below the leading entry (the first non-zero number from the left in a row) zero.

Reduced Row-Echelon Form (RREF) is a further simplification of row-echelon form in which every leading entry is 1 and is the only non-zero entry in its column. To convert the row-echelon form to RREF, you apply additional row operations:
• Ensure that each leading 1 has all zeros in its column.
• Further simplify to make all non-leading entries zero.
Once the left side of the augmented matrix is in RREF (identity matrix), the right side will be the inverse of the original matrix. In our exercise, this involves steps until we get:
\[\begin{bmatrix} 1 & 0 & 0 & 0 & | & \frac{7}{6} & \frac{2}{3} & \frac{5}{6} & -\frac{1}{2} \ \ 0 & 1 & 0 & 0 & | & \frac{1}{3} & \frac{1}{3} & -\frac{1}{3} & 0 \ \ 0 & 0 & 1 & 0 & | & -1 & 0 & -1 & 1 \ \ 0 & 0 & 0 & 1 & | & \frac{1}{2} & 0 & \frac{1}{2} & -1 d\begin{bmatrix}\].
Observe that the left side is now the identity matrix and the right side shows the inverse.

An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It plays a critical role in matrix algebra, such as in finding the inverse of a matrix. Consider an identity matrix of size 4x4:
\[\begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 d\begin{bmatrix}\]
The identity matrix acts as the matrix equivalent of the number 1 in arithmetic. When a matrix is multiplied by its inverse, the result is the identity matrix. For instance, if you multiply the original matrix from our exercise by its inverse:
\[\begin{bmatrix} 1 & -2 & 3 & 0 \ 0 & 1 & -1 & 1 \ -2 & 2 & -2 & 4 \ 0 & 2 & -3 & 1 d\begin{bmatrix} \timesd\begin{bmatrix}\frac{7}{6} & \frac{2}{3} & \frac{5}{6} & -\frac{1}{2} \ 0 & 1 & -1 & 1\frac{1}{3} & \frac{1}{3} & -\frac{1}{3} & 0 \ \ -1 & 0 & -1 & 1 \ \ \frac{1}{2} & 0 & \frac{1}{2} & -1 d\begin{bmatrix}\]
The result is the identity matrix, confirming the inverse was found correctly.