Cramer's Rule Cramer's rule is a theorem in linear algebra that relates the determinants of the numerator and denominator matrices of a partitioned linear s...
Cramer's Rule
Cramer's rule is a theorem in linear algebra that relates the determinants of the numerator and denominator matrices of a partitioned linear system to the determinants of the submatrices formed by removing the rows and columns corresponding to the pivot entries.
Notation:
(A) and (B) are (m \times n) matrices.
(A \rightleftharpoonmatrix{a_{ij}} \right|B \rightleftharpoonmatrix{b_{ij}}) is a partitioned linear system.
(A ) is invertible.
Theorem:
The determinant of the numerator matrix is equal to the determinant of the denominator matrix.
Example:
Suppose we have the following partitioned linear system:
\begin{split} 2 & 1 \\\ 3 & 4 \end{split} \begin{bmatrix} a_{11} & a_{12} \\\ a_{21} & a_{22} \\\ a_{31} & a_{32} \end{bmatrix} \begin{bmatrix} b_{11} \\\ b_{12} \\\ b_{21} \\\ b_{22} \\\ b_{31} \\\ b_{32} \end{bmatrix} = \begin{split} 0 & a_{11} \\\ 0 & a_{21} \\\ 0 & a_{31} \end{split} \end{split}
Then, by Cramer's rule, we have:
Applications:
Cramer's rule has numerous applications in linear algebra, including:
Solving linear systems of equations
Finding the determinant of a matrix
Computing eigenvalues and eigenvectors
Determining the rank of a matrix
Conclusion:
Cramer's rule is a powerful tool that provides a direct connection between the determinants of the numerator and denominator matrices of a partitioned linear system. This theorem has wide applications in linear algebra and is used to solve a wide range of problems in this field of study