Introduction to Determinants
For any square matrix of order 2, we have found a necessary and sufficient
condition for invertibility. Indeed, consider the matrix
The matrix A is invertible if and only if
 .
We called this number the determinant of A. It is clear from this,
that we would like to have a similar result for bigger matrices (meaning higher
orders). So is there a similar notion of determinant for any square matrix,
which determines whether a square matrix is invertible or not?
In order to generalize such notion to higher orders, we will need to
study the determinant and see what kind of properties it satisfies. First let us
use the following notation for the determinant
Properties of the Determinant
1.
Any matrix A and its transpose have the same determinant, meaning
This is interesting since it implies that whenever we use rows, a similar
behavior will result if we use columns. In particular we will see how row
elementary operations are helpful in finding the determinant. Therefore, we
have similar conclusions for elementary column operations.
2.
The determinant of a triangular matrix is the product of the entries on
the diagonal, that is
3.
If we interchange two rows, the determinant of the new matrix is the
opposite of the old one, that is
4.
If we multiply one row with a constant, the determinant of the new
matrix is the determinant of the old one multiplied by the constant, that is
In particular, if all the entries in one row are zero, then the determinant
is zero.
5.
If we add one row to another one multiplied by a constant, the
determinant of the new matrix is the same as the old one, that is
Note that whenever you want to replace a row by something (through
elementary operations), do not multiply the row itself by a constant.
Otherwise, you will easily make errors (due to Property 4).
6.
We have
In particular, if A is invertible (which happens if and only if
 ),
then
If A and B are similar, then
 .
Let us look at an example, to see how these properties work.
Example. Evaluate
Let us transform this matrix into a triangular one through elementary
operations. We will keep the first row and add to the second one the first
multiplied by
 .
We get
Using the Property 2, we get
Therefore, we have
which one may check easily.
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