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Determinant calculation is an essential first step in determining whether a matrix is invertible and for finding the inverse of a matrix. For 2x2 matrices, the determinant can be calculated using the formula: \(\det(A) = ad - bc\), where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix \(A = \left(\begin{array}{cc}a & b \ c & d\end{array}\right)\).

If the determinant of a matrix is zero, the matrix does not have an inverse and is referred to as singular. Conversely, if the determinant is non-zero, the matrix is nonsingular and has an inverse. For example, the determinants of matrices in the exercise highlight their invertibility: matrix (e) has a determinant of zero and is not invertible, while the others have non-zero determinants and therefore possess inverses. Understanding how to calculate the determinant is crucial for these determinations.

Once the determinant of a 2x2 matrix is found to be non-zero, its inverse can be computed using a specific formula. The inverse of a 2x2 matrix \(A = \left(\begin{array}{cc}a & b \ c & d\end{array}\right)\), if it exists, is given by \(A^{-1} = \frac{1}{\det(A)} \left(\begin{array}{cc}d & -b \ -c & a\end{array}\right)\).

This formula involves swapping the elements of the main diagonal, changing the signs of the off-diagonal elements, and then scaling the resultant matrix by the reciprocal of the determinant. For instance, in matrix (c) from the exercise, once we have found the determinant to be \-10\, we swap \(1\) and \(2\), change the signs of \(4\) and \(3\), and multiply each element by \-0.1\ to obtain the inverse.

To determine the invertibility of a matrix, one must consider its determinant. A square matrix is invertible, also known as non-singular, only if its determinant is non-zero. This is because only non-singular matrices have an associated multiplicative inverse, which when multiplied by the matrix yields the identity matrix.

For exercises like the ones provided, identifying matrices with zero determinants is a quick way to determine non-invertibility. For example, matrix (e) with a zero determinant is not invertible. It's important to note that non-square matrices cannot be inverted, as the concept of an inverse only applies to square matrices. However, there are generalizations such as pseudoinverses for non-square matrices. In practice, a matrix's invertibility is also tied closely to the concept of rank and the solutions to linear systems associated with the matrix.