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The determinant of a matrix plays a crucial role in various aspects of linear algebra, including determining whether a matrix is invertible. For a 2x2 matrix like \[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, \]the determinant is calculated using the formula:\[ \text{det}(A) = ad - bc. \]In the example matrix \(A = \begin{bmatrix} -1 & 4 \ 5 & 1 \end{bmatrix},\)substitute the values where \(a = -1\), \(b = 4\), \(c = 5\), and \(d = 1\).
This gives us:\[ \text{det}(A) = (-1)(1) - (4)(5) = -1 - 20 = -21. \]The determinant not only helps find the inverse but also checks if a matrix is invertible. A non-zero determinant indicates the matrix can indeed be inverted, as shown here.

The adjugate of a 2x2 matrix is used when finding the matrix inverse, especially in manual calculations. It's derived by swapping the diagonal elements and changing the signs of the off-diagonal elements. For a general 2x2 matrix:\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix}, \]the adjugate is:\[ \text{adj}(A) = \begin{bmatrix} d & -b \ -c & a \end{bmatrix}. \]Applying this to our specific example matrix \(A = \begin{bmatrix} -1 & 4 \ 5 & 1 \end{bmatrix}\),we find the adjugate matrix:\[ \text{adj}(A) = \begin{bmatrix} 1 & -4 \ -5 & -1 \end{bmatrix}. \]Using the adjugate is a key step in ultimately determining the inverse of a matrix.

Understanding when a matrix is invertible is fundamental in linear algebra. A square matrix is said to be invertible—or non-singular—if its determinant is not equal to zero. This criterion ensures that a unique inverse matrix exists. The inverse is utilized in solving linear equations and transforming systems.
For our example matrix \(A = \begin{bmatrix} -1 & 4 \ 5 & 1 \end{bmatrix}\),we calculated that \(\text{det}(A) = -21\),which is non-zero. This confirms that matrix \(A\) is indeed invertible.
An invertible matrix allows for meaningful computations such as finding solutions to linear systems, which is often represented as\[ A \mathbf{x} = \mathbf{b}. \]Knowing your matrix is invertible gives the green light for calculating other important characteristics in linear transformations.