WEBVTT
00:00:00.637 --> 00:00:07.987
Calculate the determinant of 𝐴 when 𝐴 equals three, zero, negative one, zero, one, zero, two, two, four.
00:00:08.857 --> 00:00:15.387
To make things easier for ourselves, we’ll identify the row or column that contains the most number of entries which are zero.
00:00:15.957 --> 00:00:19.637
For this matrix, that’s going to be the second row of 𝐴.
00:00:20.277 --> 00:00:25.807
Let’s remind ourselves of the general formula for finding the determinant of an 𝑛-by-𝑛 matrix.
00:00:26.567 --> 00:00:27.747
Here is the formula.
00:00:28.067 --> 00:00:32.617
Remember that 𝑖 represents the row number and 𝑗 represents the column number.
00:00:33.477 --> 00:00:37.077
So as we have three columns, 𝑗 runs from one to three.
00:00:37.487 --> 00:00:40.907
And we’re expanding along the second row, so 𝑖 equals two.
00:00:41.447 --> 00:00:44.037
So this is the formula that we’re going to be using.
00:00:44.747 --> 00:00:51.307
The entries 𝑎 two one, 𝑎 two two, and 𝑎 two three are zero, one, and zero, respectively.
00:00:51.777 --> 00:00:59.327
Because two of these entries are zero, we find that both of these terms are going to be zero because they’re both being multiplied by zero.
00:00:59.807 --> 00:01:02.977
So there’s actually only one term that we need to calculate here.
00:01:03.717 --> 00:01:06.967
Let’s begin by calculating the matrix minor 𝐴 two two.
00:01:07.597 --> 00:01:12.857
This is going to be the two-by-two matrix we get by removing the second row and the second column.
00:01:13.527 --> 00:01:19.327
So we see that the matrix minor 𝐴 two two is three, negative one, two, four.
00:01:20.047 --> 00:01:23.487
But what we actually need for our formula is its determinant.
00:01:23.917 --> 00:01:28.947
So we calculate this in the usual way for finding determinants of a two-by-two matrix.
00:01:29.567 --> 00:01:33.767
This gives us three times four minus negative one times two.
00:01:34.337 --> 00:01:38.707
This gives us 12 minus negative two, which gives us 14.
00:01:39.657 --> 00:01:44.187
The final thing we can do is calculate the value of negative one to the power of two add two.
00:01:44.627 --> 00:01:51.187
As this is negative one to the fourth power, which is an even power, this is going to give us one.
00:01:51.877 --> 00:01:56.317
So we calculate the determinant of 𝐴 to be one times one times 14.
00:01:57.017 --> 00:01:59.447
And this, of course, just gives us 14.
00:02:00.287 --> 00:02:09.417
So carefully selecting the row or column that you choose when you’re finding the determinant of a matrix can really reduce the amount of calculations necessary.