Basic Examples (1) 

Form the adjugate of a matrix of dimension 2:

In[1]:=

Out[1]=

Scope (8) 

The adjugate is a square matrix of the same dimension:

In[2]:=

In[3]:=

Out[3]=

In[4]:=

Out[4]=

The product of the adjugate with the matrix is a diagonal matrix with the same values on the diagonal:

In[5]:=

Out[5]=

The values on the diagonal are the determinant of the matrix:

In[6]:=

Out[6]=

When a matrix is invertible, the adjugate divided by the determinant gives the inverse:

In[7]:=

Out[7]=

The adjugate is defined for symbolic as well as numeric matrices:

In[8]:=

Out[8]=

A square matrix need not be invertible:

In[9]:=

This matrix is not invertible because it does not have full rank:

In[10]:=

Out[10]=

The adjugate is defined even when the inverse does not exist:

In[11]:=

Out[11]=

Properties and Relations (2) 

The adjugate satisfies a simple identity:

In[12]:=

In[13]:=

Out[13]=

The adjugate and the matrix of minors are the same up to reordering and signs:

In[14]:=

Out[15]=

In[16]:=

Out[16]=

In[17]:=

Out[17]=

The adjugate and the matrix of cofactors (implemented as the resource function CofactorMatrix) are transposes of one another:

In[18]:=

Out[18]=