5. Determinants

The determinant of a square matrix is a real number that gives us valuable information about the matrix. Its definition is cumbersome, so it is in a special section at the end of this chapter. You will see some of the uses of the determinant in later chapters. For now, let's find out how to compute the determinant of a matrix so that we can use it later. The symbols det(A) and |A| represent the determinant of A. In this case, the straight bars do NOT mean absolute value; they represent the determinant of the matrix. The determinant of a 1 by 1 matrix is simply the element of the matrix. If A is the 2 by 2 matrix, then det(A) = ad - bc is found this way:

You may remember ad-bc from the last chapter. If det(A) ≠ 0, then the inverse of the 2 by 2 matrix, A, is

,

which can also be written as

We have already found the determinant for a 2 by 2 matrix, so let's look at the 3 by 3 matrix

.

In the 2 by 2 case, we subtracted products of the diagonals from each other beginning with the main diagonal. If we do that with the 3 by 3 case, we will be leaving out 4 of the 9 numbers. Let's rewrite the first two columns of A so that each number of the original matrix falls on a diagonal. It looks like this:

If we only use the diagonals that have 3 numbers on them, we will be using every number that was in our original matrix the same number of times. The diagonals towards the upper left give us 6*2*0 = 0 and 1*3*2 = 6 and 3*4*4 = 48. If we add these together, we get 54. Now let's look at the diagonals towards the upper right. They give us 1*2*4 = 8 and 3*4*0 = 0 and 6*4*2 = 48. If we add these together, we get 56. Let's subtract the upper right from the upper left like we did with the 2 by 2 case. We obtain | A| = -2. The method of finding the determinant for the 3 by 3 matrix that we showed you is called "repeat the columns" for obvious reasons. Look at the numbers you used and where they came from. See if you can find a way to determine these same numbers without writing down the columns each time. It could look like this:

One of our favorite high school teachers likes to call this the "spaghetti method." It is also called the "basket weave method".