Main Points:
- Adjoint and multiplicative inverse of a square matrix:
- The adjoint of a matrix is a matrix that is obtained by taking the transpose of the matrix of cofactors of the original matrix.
- The adjoint matrix is used to calculate the inverse of a matrix.
- The multiplicative inverse of a matrix is a matrix that, when multiplied by the original matrix, gives the identity matrix.
- A square matrix is invertible if and only if its determinant is nonzero.
- The inverse of a matrix can be used to solve systems of linear equations, find the coefficients of a polynomial, and perform other mathematical operations.
Formulas to remember
- Adjoint of A: $$(adj\ A)_{ij} = (-1)^{i+j} M_{ji}$$
- Multiplicative Inverse of A: $$A^{-1} = \frac{1}{|A|} adj\ A$$