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The determinant of a matrix is a special number that provides useful properties for the matrix. Specifically, it helps in determining whether a matrix has an inverse. For a 3x3 matrix like: \(A = \begin{pmatrix} 2 & 3 & 3 \ 1 & 4 & 3 \ 1 & 3 & 4 \end{pmatrix}\) The determinant is found by breaking the matrix into smaller 2x2 submatrices. We follow this formula for the determinant of a 3x3 matrix: \({\text{det}}(A) = a_{11} \begin{vmatrix} a_{22} & a_{23} \ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \ a_{31} & a_{32} \end{vmatrix}\) We then calculate the minors and perform the arithmetic: \({\text{det}}(A) = 2 \cdot 7 - 3 \cdot 1 + 3 \cdot (-1) = 14 - 3 - 3 = 8\) Because the determinant is non-zero (8), matrix \(A\) has an inverse.

To find the inverse of a matrix, we need to ensure its determinant is not zero. We've established that the determinant of our given matrix is 8, so we can proceed with finding the inverse. The inverse of a matrix \(A\) is denoted as \(A^{-1}\) and is found using: \(A^{-1} = \frac{1}{{\text{det}}(A)} C^T\) where \(C\) is the cofactor matrix and \(C^T\) represents its transpose. This means each element of the transposed cofactor matrix must be divided by the determinant. The resulting matrix will be the inverse we are looking for.

The cofactor matrix is crucial for finding the inverse of a matrix. To calculate the cofactor matrix, we need to find minors for each element of the original matrix, and apply a sign change based on the element's position using the formula \(((-1)^{i+j}) \times {\text{minor}}_{ij}\). For our 3x3 matrix \(A\), we compute: \(C_{11} = \begin{vmatrix} 4 & 3 \ 3 & 4 \end{vmatrix} = 7\) \(C_{12} = -((1 \times 4) - (3 \times 3)) = -1\) \(C_{13} = ((1 \times 3) - (4 \times 1)) = -1\) Similarly, we calculate all the other cofactors and arrange them in the cofactor matrix \(C\).

The transpose of a matrix is obtained by swapping its rows and columns. For example, if matrix \(A\) is: \(A = \begin{pmatrix} 2 & 3 & 3 \ 1 & 4 & 3 \ 1 & 3 & 4 \end{pmatrix}\) The transpose of the cofactor matrix, denoted as \(C^T\), involves taking each element from position \((i, j)\) in the cofactor matrix \(C\) and placing it at position \((j, i)\). After finding the cofactor matrix elements and arranging them, the transpose process will give us the final arrangement needed for further calculations. Finally, divide each element of the transposed cofactor matrix by the determinant to find the inverse.