Some properties of Determinants

 

·

The value of the determinant of a matrix doesn't change if we transpose this matrix (change rows to columns)

· a is a scalar, A is n´ n matrix

If we multiply a scalar to a matrix A, then the value of the determinant will change by a factor !

· If an entire row or an entire column of A contains only zero's, then

This makes sense, since we are free to choose by which row or column we will expand the determinant. If we choose the one containing only zero's, the result of course will be zero. And - it must be zero for all other possible expansions, too.

·

If two determinants differ by just one column, we can add them together by just adding up these two columns. For example:

 

How do determinants behave in elementary row operations?

· Interchange of two rows:

The sign of the determinant will change if you interchange two rows - this has to do with the checkerboard pattern of the coefficients !

· Corresponding entries in two rows are proportional

If the entries of two rows turn out to be proportional to each other we are able to eliminate one of these row entirely during Gauss elimination: all entries of one row eventually will become zero. Since we can choose this particular row as the one we expand the determinant by the result will become zero!

· Adding a multiple of a row/column to another

All other elementary row operations will not affect the value of the determinant!

We can use Gauss elimination to reduce a determinant to a triangular form….

Benefit: After this, we only have to multiply the diagonal elements with each other.

This turns out to be an enormous help to us: instead of calculating endlessly one minor after the other, we now have the alternative to simply make our determinant upper triangular using the same row operations we used during Gauss elimination.

However, there is one (but only one) thing we have to pay attention to: If we exchange two rows with each other, the sign of the determinant will switch. We must keep track on how often we interchange rows!

Let us try out what we just learned. Let us use Gauss elimination in order to obtain the following determinant:

gives

gives

Is this the right result?

Let us verify it:

Same result!

In this example it was sort of hard to tell which method is easier. But imagine some 6th order determinant containing almost no zero's…..

Try to obtain

using both methods…..

There are more interesting properties of determinants …

Determinant of a Product of Matrices:

A, B are matrices, then

This is kind of surprising since we know that – in general – we cannot reverse the order of matrix multiplication:

Let us demonstrate this new feature:

Demonstration:

and also

Therefore, . Yes, of course: A determinant can be zero even if not a single entry of the determinant itself is zero!

 

How about the matrix products?

Therefore, indeed,

Let’s do this with another example – one, where the determinants do not equal to zero (just to make sure…). Let consider

Then,

and, for the products we get

and

It seems to work!

 

Rank in terms of determinants

Here is another conclusion that might become handy:

For a matrix A:

Û exists (or )

If the value of a nth order determinant is not equal zero, then the rank of the associated matrix must be n.

This makes perfect sense: Since we can use Gauss elimination in order to simplify the calculation of our determinant eventually an entire row of the determinant has to be filled with zero’s in order to let the determinant become zero. But then the rank of the associated matrix would be smaller than n – it would be n-1 the most.

Later on we will see that this is also an indicator if the inverse of a matrix does exist….

 

 

Copyrights 1999, 2000 by Peter Dragovitsch and Ben A. Fusaro