Sometimes it is possible to transform (modify) a determinant in such a way that it is easier to evaluate, without changing its value. One of the tools we have at our disposal to do this is called "row reduction." 

Reduction Rule #1
If we add any multiple of one row to any other row, or if we add any multiple of one column to any other column, we don't alter the value of the determinant.

How can we use this rule to help us evaluate a determinant? Well, take a look at the following determinant:

Watch what happens if we subtract the first row from the second row; we'll obtain the following determinant (which has the same value!)

Now we have a column with a zero in it, which makes it easier to evaluate by minors. But that's not all; we could subtract twice the first row from the third row, and now we'll have:

That's even better, because we now have a column with two zeroes, making it even easier to evaluate.

Reduction Rule #2
If we can manipulate our determinant in such a way that all the values above (or below) the main diagonal are zeroes, the value of the determinant is just the product of the values in the diagonal.

So look what happens if we add the third column to the second column:

Now, since we have nothing but zeroes under the main diagonal, we can just multiply these elements, and we have the value of the determinant: (1)(1)(-4) = -4

Reduction Rule #3
If you interchange any two rows, or any two columns of a determinant, you have multiplied its value by -1.

Look at the following determinant:

We could evaluate this determinant as it is, but if we swapped the second and third rows, we would have this:

= -, and this determinant has all zeroes below the main diagonal, so we can quickly use our second reduction rule to evaluate it: -5(2)(8) = -80.

Reduction Rule #4
If two rows (or two columns) of a determinant are identical, the value of the determinant is zero.

Consider the following determinant:

If we subtract the first row from the second row, we obtain:

Since the first and second rows are now equal, we know the value of the determinant is zero.

Reduction Rule #5
If any row or column has only zeroes, the value of the determinant is zero.

This makes sense, doesn't it? If you expanded around that row/column, you'd end up multiplying all your determinants by zero!

In the previous example, if we had subtracted twice the first row from the second row, we would have obtained:

, which has a row of zeroes, confirming that the value of the determinant is zero.

1.

Which reduction rule most easily gives us the value of this determinant?

2.

What would you do to this determinant to get all zeroes below the main diagonal?

3.

Which rule should we use to evaluate this determinant?

4.

Which rule should we use to evaluate this determinant?