# Determinants of Matrices of Higher Order

As we said before, the idea is to assume that previous properties satisfied by the determinant of matrices of order 2, are still valid in general. In other words, we assume:

1.
Any matrix A and its transpose have the same determinant, meaning

2.
The determinant of a triangular matrix is the product of the entries on the diagonal.
3.
If we interchange two rows, the determinant of the new matrix is the opposite of the old one.
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.
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.
6.
We have

In particular, if A is invertible (which happens if and only if ), then

So let us see how this works in case of a matrix of order 4.

Example. Evaluate

We have

If we subtract every row multiplied by the appropriate number from the first row, we get

We do not touch the first row and work with the other rows. We interchange the second with the third to get

If we subtract every row multiplied by the appropriate number from the second row, we get

Using previous properties, we have

If we multiply the third row by 13 and add it to the fourth, we get

which is equal to 3. Putting all the numbers together, we get

These calculations seem to be rather lengthy. We will see later on that a general formula for the determinant does exist.

Example. Evaluate

In this example, we will not give the details of the elementary operations. We have

Example. Evaluate

We have

General Formula for the Determinant Let A be a square matrix of order n. Write A = (aij), where aij is the entry on the row number i and the column number j, for and . For any i and j, set Aij (called the cofactors) to be the determinant of the square matrix of order (n-1) obtained from A by removing the row number i and the column number j multiplied by (-1)i+j. We have

for any fixed i, and

for any fixed j. In other words, we have two type of formulas: along a row (number i) or along a column (number j). Any row or any column will do. The trick is to use a row or a column which has a lot of zeros.
In particular, we have along the rows

or

or

As an exercise write the formulas along the columns.

Example. Evaluate

We will use the general formula along the third row. We have

Which technique to evaluate a determinant is easier ? The answer depends on the person who is evaluating the determinant. Some like the elementary row operations and some like the general formula. All that matters is to get the correct answer.

Note that all of the above properties are still valid in the general case. Also you should remember that the concept of a determinant only exists for square matrices.

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Author: M.A. Khamsi