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Calculating the determinant is essential to find out whether a matrix is invertible. The determinant is a special number calculated from a square matrix. If the determinant is non-zero, the matrix can be inverted. For a 3x3 matrix, the determinant is given by the formula:

• Multiply the diagonal elements from top left to bottom right and subtract the product of the other diagonals.
• Determinant for matrix \[ A = \begin{bmatrix} 1 & 0 & 0 \ 1 & 1 & 0 \ 0 & 1 & 1 \end{bmatrix} \] is calculated as follows:
• \[ \text{det}(A) = 1(1 \cdot 1 - 0 \cdot 1) - 0(1 \cdot 1 - 0 \cdot 0) + 0(1 \cdot 1 - 1 \cdot 0) = 1 \cdot 1 = 1.\]

This result shows us that the determinant is 1, which confirms that the matrix is invertible.

Row operations are steps we use to simplify matrices. They help transform the matrix into a form that's easier to work with. Important types of row operations include:

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

In matrix inversion, the goal is to use these operations to change the left side of the augmented matrix into an identity matrix. By doing so, the operations on the right side reveal the inverse matrix. These operations are purposely reversible, ensuring the overall matrix does not change its inherent properties.
In the example provided, specific operations such as \(R_2 - R_1 \rightarrow R_2\) and \(R_3 - R_2 \rightarrow R_3\) are applied to reach a desired row form. This moves the original matrix's elements toward forming the identity on its side.

An augmented matrix is used when we want to apply row operations to find solutions or invert matrices. It combines the original matrix with another, aiding in manipulation and calculation. To form an augmented matrix:

• Place the original matrix, \(A\), on the left.
• Position the identity matrix, \(I\), on the right to get \([A | I]\).

The idea is to use row operations so that the left half transforms into the identity matrix while the right half changes into the inverse of \(A\). In our context, this is represented as:\[\left[ \begin{array}{ccc|ccc} 1 & 0 & 0 & 1 & 0 & 0 \1 & 1 & 0 & 0 & 1 & 0 \0 & 1 & 1 & 0 & 0 & 1 \end{array} \right]\]
This setup aids in simultaneously transforming an equation system and directly finding the inverse.

The identity matrix is an important concept in linear algebra. It's a square matrix where all the elements on the main diagonal are ones, and all other elements are zeros. In matrix terms, it's the equivalent of the number 1 in basic arithmetic.

• This matrix does not affect other matrices when involved in multiplication.
• It's denoted typically as \(I\) and sized appropriately to fit the context, such as \(I_3\) for a 3x3 matrix.

When finding an inverse, achieving the identity matrix on one side of an augmented matrix means the right side has transformed into the inverse matrix. Through row operations, transforming the left side of the augmented matrix to the identity allows for the extraction of the inverse matrix from the right side. For instance, reaching:\[\left[ \begin{array}{ccc} 1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \end{array} \right]\]means success in retrieving \(A^{-1}\) from the augmented part.