Unlike matrix-matrix multiplication, there exists no direct analog for matrix division. However, if we recall that division of quantity 
by quantity 
is simply multiplication of 
by 
's inverse (
) we can similarly define a matrix inverse with which to perform division. Using the determinant, we can define the inverse of a 
matrix 
in the following way:
by quantity is simply multiplication of by 's inverse () we can similarly define a matrix inverse with which to perform division. Using the determinant, we can define the inverse of a matrix in the following way:Excercise 3-11.
Find the inverse of the following matrix: 
And just as 
, similarly we have 
where 
is of course the identity matrix.
Not all matrices are invertible. A square matrix is the only type of matrix that is invertible, though one can also define a left or right inverse for a non-square matrix (not covered in this book). Not all square matrices are invertible. As can be seen from the equation for a
inverse, if the determinant equals 0, the inverse is undefined. Thus we can say the following holds: 
Note that the two-headed arrow is a double imply or if and only if (
) statement. This means that if either the right or left hand side of the relation is true then the other side must also be true.
The determinant can be used to find the inverse of higher dimensional matrices, but the approach is cumbersome. The subsection on Gauss-Jordan Elimination describes a technique which is more tractable.
, similarly we have where is of course the identity matrix.Not all matrices are invertible. A square matrix is the only type of matrix that is invertible, though one can also define a left or right inverse for a non-square matrix (not covered in this book). Not all square matrices are invertible. As can be seen from the equation for a
inverse, if the determinant equals 0, the inverse is undefined. Thus we can say the following holds: ) statement. This means that if either the right or left hand side of the relation is true then the other side must also be true.The determinant can be used to find the inverse of higher dimensional matrices, but the approach is cumbersome. The subsection on Gauss-Jordan Elimination describes a technique which is more tractable.