Video: 3 × 3 Determinants

Calculate |𝐴| when 𝐴 = [3, 0, −1 and 0, 1, 0 and 2, 2, 4].

02:10

Video Transcript

Calculate the determinant of 𝐴 when 𝐴 equals three, zero, negative one, zero, one, zero, two, two, four.

To make things easier for ourselves, we’ll identify the row or column that contains the most number of entries which are zero. For this matrix, that’s going to be the second row of 𝐴. Let’s remind ourselves of the general formula for finding the determinant of an 𝑛-by-𝑛 matrix. Here is the formula. Remember that 𝑖 represents the row number and 𝑗 represents the column number. So as we have three columns, 𝑗 runs from one to three. And we’re expanding along the second row, so 𝑖 equals two. So this is the formula that we’re going to be using.

The entries 𝑎 two one, 𝑎 two two, and 𝑎 two three are zero, one, and zero, respectively. 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. So there’s actually only one term that we need to calculate here. Let’s begin by calculating the matrix minor 𝐴 two two. This is going to be the two-by-two matrix we get by removing the second row and the second column. So we see that the matrix minor 𝐴 two two is three, negative one, two, four. But what we actually need for our formula is its determinant. So we calculate this in the usual way for finding determinants of a two-by-two matrix. This gives us three times four minus negative one times two. This gives us 12 minus negative two, which gives us 14.

The final thing we can do is calculate the value of negative one to the power of two add two. As this is negative one to the fourth power, which is an even power, this is going to give us one. So we calculate the determinant of 𝐴 to be one times one times 14. And this, of course, just gives us 14. 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.

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