Adjugate matrix

In linear algebra, the adjugate matrix $\text{adj}(A)$ (also adjoint matrix) of an invertible $n\times n$ matrix $A$ is an $n\times n$ matrix, whose $ij$th entry is given by:

$$(-1{)}^{i+j}\det (A_{ji})$$

This is used in another definition of the inverse matrix:

$$A^{-1}=\frac{1}{\det A}\text{adj}(A)$$

In MATLAB, we can use adjoint(A).

Computation

As mentioned above, each entry in the adjugate matrix is given by:

$$(-1{)}^{i+j}\det (A_{ji})$$

i.e., each entry is computed by calculating the cofactor of the element. We essentially calculate the determinant of the resulting matrix without the element’s row/column.

Then, to get the final matrix, we take its transpose. Don’t forget this!