Calculating determinants by using the formal definition is generally a huge amount of work - there are better ways. But there are a few special cases for which it's relatively easy to calculate determinants using elementary products. For example, suppose a determinant has a row or column of zeroes. Since every elementary product must contain one number from that row or column, all are 0, so the value of the determinant is 0.

The "diagonals" method. If the matrix is small (2 x 2 or 3 x 3), you can find the signs of the elementary products by a type of "down-right is +, down-left is – " rule. Here's how that works.

 

There's one elementary product for which the sign is always easy to determine: the product of the numbers down the main diagonal. These numbers all have the same column number as row number, so zero swaps are necessary to transform the list of row numbers into the list of column numbers. Since 0 is an even number, the sign is +.

This observation makes it easy to calculate the determinant of a diagonal matrix: its only non-zero elementary product is the product of the numbers on its diagonal. Since that elementary product has a positive sign, the determinant of a diagonal matrix is the product of the numbers along its main diagonal – for example,

.

We can extend this idea to triangular matrices as well. Recall that a square matrix is upper triangular if all entries below its main diagonal are 0, and lower triangular if all entries above its main diagonal are 0. Here's an example of an upper triangular matrix.

 

Suppose we know the determinant of a matrix and want to find the determinant of its transpose. The transpose has the same elementary products as the original matrix, so we just have to worry about the signs attached to the elementary products.

Recall what we did for the original matrix: for each elementary product, we listed the row and column positions of each entry involved and counted the number of swap needed to transform the list of column positions into the list of row positions. Here's the example again.

Entry	3	5	–4	2
Row position	1	2	3	4
Column position	3	4	2	1
1.   3 ↔ 1  produces	1	4	2	3
2.   4 ↔ 2  produces	1	2	4	3
3.   3 ↔ 4  produces	1	2	3	4
Total number of swaps: 3

We could just as well have swapped column positions to transform the list of column positions into the list of row positions: you use exactly the same swaps, but in the reverse order.

Entry	3	5	–4	2
Column position	3	4	2	1
Row position	1	2	3	4
1.   3 ↔ 4  produces	1	2	4	3
2.   4 ↔ 2  produces	1	4	2	3
3.   3 ↔ 1  produces	3	4	2	1
Total number of swaps: 3

There are the same number of swaps, so it really doesn't matter if we start with row positions or column positions. This works in general, for any number of swaps and for any square matrix.

But then, finding the signs for the elementary products of the transpose matrix by starting with columns is the same as finding the appropriate signs for the elementary products of the original matrix by starting with rows. Each elementary product has the same signs as before. So the matrix and its transpose have the same signed elementary products, and thus the same determinant. We conclude that the determinant of a matrix equals the determinant of its transpose: for any square matrix A, det(AT) = det(A).

One last useful rule. Suppose the entries of some row of a determinant all consist of sums, for example

.

 

When you calculate this determinant, each elementary product contains exactly one entry from this row, and so contains a sum:

(a + p)ei + (b + q)fg + (c + r)dh - (c + r)eg - (a + p)fh - (b + q)di.

Expand each term and then collect together all terms which came from the first number in one of the original sums, and all the terms which came form the second number in one of the original sums:

{aei + bfg + cdh - ceg - afh - bdi} + {pei + qfg + rdh - reg - pfh - qdi}.

The result is two determinants identical except for one row - one determinant has the first numbers from the sums in that row and the other has the second numbers form the sums in that row.

.

This calculation generalizes to any size determinant and can be used with columns as well.

For a 2 x 2 determinant, the elementary product on the down-right diagonal has a + sign attached, and the elementary product on the down-left diagonal has a – sign attached.

For a 3 x 3 determinant, write ghosts of the first two columns on the right. The elementary products along the down-right diagonals are assigned plus signs and the ones along the down-left diagonals are assigned minus signs.

These rules give simple shortcuts for small determinants that are definitely worth learning. But the diagonals rule does not work for larger matrices – it doesn't even give all the elementary products, let alone their correct signs. Look again at the 4 x 4 example on the previous page and note that only 8 of the 24 elementary products lie on a down-right or down-left diagonal.

The determinant of a triangular matrix is the product of the numbers along its main diagonal.

Proof: Suppose first the matrix is upper triangular. Most elementary products contain at least one zero and so are 0. Search for ways you can get a non-zero elementary product.

You must take a number from each column.

For column 1, the only possibility is the number in row 1.

For column 2, you can't choose the number in row 1, since you've already used a number from row 1, so you must choose the second number in row 2, since all other numbers in that column are 0.

And so on: the only way you can possibly get a non-zero elementary product is to take the product of the numbers down the main diagonal. (And if any of those is 0, you can't get any non-zero elementary products.) The sign of this elementary product is +, so the determinant is the product of the numbers down its main diagonal.

For a lower triangular matrix, the same basic idea works; just look at which rows you can choose your numbers from.