To do:

Practice calculations with adjoints and inverses.

Investigate the values of the determinant of the adjoint in relation to the determinant of the original matrix.

When is the adjoint of a matrix equal to the zero matrix? Does the matrix have to be zero?

What is the relation between the adjoint of a product and the product of the adjoints?
You can try small samples and then make a guess. Also, you may use the known results for determinants and inverses.

You know that the inverse of an inverse gives the original matrix. What about the adjoint of an adjoint?

Given two row vectors , we can form a matrix .
Define a new vector , where is the determinant obtained by dropping the -th column and multiplying it by .
Calculate the dot product of a row vector with and write it as a suitable determinant.
Interpret the entries as entries of the adjoint for a suitable matrix.
Use determinant to prove that are both perpendicular to . This is the well known cross product of two vectors and it is defined only in three dimensions.

Explain why , gives the equation of a plane thru ( ) and parallel to the vectors . You need to connect the geometry with the theory of determinants. The idea is to make the argument without expanding!

More generally, given vectors with components each, we can form a determinant by augmenting (or stacking) them. The determinant gives a natural formula for a generalized parallelopiped formed by the vectors. For this is easy to check, in higher dimensions it is a definition. The determinant vanishes when the configuration is degenerate.
This means that for , the two vectors lie along the same line.
For , it means that the three vectors lie in a plane and so the parallelopiped is squashed down.
In general, it says that they lie in a smaller than dimensional space.

We have been using the word dimension freely. We study it in detail when we get to chapter 3.