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 𝑚-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.