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The determinant of a matrix is a special number that can be calculated from its elements. This number provides important information about the matrix, such as whether the matrix is invertible. For a 3x3 matrix, the determinant is computed using the formula:

\[ \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \]
To calculate the determinant, follow these steps:

• Identify elements from the matrix (let's denote them as \( a, b, c, d, e, f, g, h, i \)).
• Substitute these elements into the formula.
• Simplify the expression to find the determinant.

For our matrix \( A \), the determinant is calculated as:
\[ \text{det}(A) = (-1)((-2) \cdot 2 - 0 \cdot 1) - 0(0 \cdot 2 - 0 \cdot -1) + (-1)(0 \cdot 1 - (-2) \cdot -1) \]
This simplifies to \( 2 \). Since the determinant is non-zero, the matrix is invertible.

A matrix cofactor is a crucial concept for finding the inverse of a matrix. Cofactors are the signed minors of a matrix element; they are used to build the cofactor matrix, an essential step in calculating the inverse.

To find the cofactor for an element in a matrix:

• Delete the row and column of the element within the matrix. This creates a smaller 2x2 matrix called a minor.
• Calculate the determinant of this minor.
• Apply a sign to the determinant, using the pattern \((-1)^{i+j}\) based on the element's position \((i,j)\) within the matrix.

Cofactors, arranged properly, will help you build the cofactor matrix. This matrix plays a key role in determining the adjugate matrix.

The adjugate, or adjoint, matrix is critical when finding the inverse of a matrix. It is the transpose of the cofactor matrix, which rearranges the positions of cofactor elements.

The steps to obtain the adjugate matrix:

• Calculate the cofactor matrix for your original matrix.
• Transpose the cofactor matrix by swapping its rows and columns.

In our context of the 3x3 matrix, the cofactor matrix was:\[\begin{bmatrix}4 & 0 & 0 \-1 & -3 & 1 \2 & 1 & 2\end{bmatrix}\]
Upon transposing, we rearrange rows into columns:\[\begin{bmatrix}4 & -1 & 2 \0 & -3 & 1 \0 & 1 & 2\end{bmatrix}\]The resulting matrix is the adjugate and is used in computing the inverse of the original matrix.

To find the inverse of a 3x3 matrix, follow a precise procedure. The inverse of a matrix \( A \) is denoted as \( A^{-1} \), and it exists only if the determinant of the matrix is not zero.

The formula to compute the inverse is:\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A) \]where \( \text{Adj}(A) \) is the adjugate matrix. Thus, you must compute:

• The determinant of the matrix \( A \).
• The cofactor matrix, and its transpose to determine the adjugate matrix.

Once you have these, substitute into the formula and perform scalar multiplication with each element of the adjugate matrix.

For the matrix \( A \) provided, we calculated:\[A^{-1} = \frac{1}{2} \cdot \begin{bmatrix}4 & -1 & 2 \0 & -3 & 1 \0 & 1 & 2\end{bmatrix}\]Simplifying gives the inverse:\[\begin{bmatrix}2 & -0.5 & 1 \0 & -1.5 & 0.5 \0 & 0.5 & 1\end{bmatrix}\]The inverse matrix is valid if multiplying it with the original matrix \( A \) returns the identity matrix, confirming our computations.