The Adjugate of a Matrix and Invertibility
We define the adjugate matrix of an nxn square matrix. This matrix is formed by constructing all principal minors of order one and considering the corresponding cofactor. These elements are arranged in another nxn square matrix according to the row and column which were deleted, and then the matrix is transposed. We prove that when the original matrix is invertible, then the adjugate divided by the determinant of the original matrix is the inverse. We will see if further videos how the adjugate plays an essential role in differentiating matrices which depend on a parameter.
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