The Cramer's rule is a theorem in linear algebra that provides a method for finding the determinant of a square matrix using determinants of smaller submatrices...
The Cramer's rule is a theorem in linear algebra that provides a method for finding the determinant of a square matrix using determinants of smaller submatrices. The rule works by utilizing the values of determinants of particular submatrices to compute the determinant of the original matrix.
For a square matrix (A) with dimensions (m \times m), the Cramer's rule states that the determinant of (A) can be calculated as follows:
where (a_{ij}) denotes the element of the (i\text{-th row} \text{ and } j\text{-th column}) of the matrix.
The Cramer's rule can be applied to any (m \times m) matrix, but it is particularly useful when (m=2). In this case, the determinant can be calculated directly from the values of determinants of smaller submatrices.
For instance, the determinant of the matrix (A) can be found as follows: