Find, in its simplest form, an expression for the determinant.
Here’s our three-by three-matrix. To find the determinate in this three-by-three matrix, we’ll use the first row. The first row the first column will be positive, first row second column negative, and first row third column positive. To find the determinant, we’re gonna break this up into three smaller determinants. Starting with nine minus two 𝑘, we want to take this value and multiply it by the smaller matrix created by removing both the row and the column that nine minus two 𝑘 is located in.
We’ll multiply positive nine minus two 𝑘 times the determinant of the two-by-two matrix created here. And this will be our first term. Our second term will be negative negative two 𝑚 multiplied by the two-by-two matrix we create when we remove the row and the column that negative two 𝑚 is found in. The two-by-two matrix would look like this: negative two 𝑘, seven 𝑛, negative two 𝑘, nine plus seven 𝑛. This will be our second term.
Following in the same pattern, we’ll take seven 𝑛 and multiply it by the two-by-two matrix created when you remove the column and the row that seven 𝑛 is located in, which will look like this and would be our third term. Now we have some major algebra to do. I’m going to bring down the part that comprises the first term. And for now, we’ll ignore the second and third term.
Nine minus two 𝑘 is what we start with, but remember we need the determinant of this two-by-two matrix. And that means multiplying negative one minus two 𝑚 times nine plus seven 𝑛 and then subtracting that value from seven 𝑛 times negative two 𝑚. And all of this will solve for our first term. Looking inside the brackets, we need to multiply negative one minus two 𝑚 times nine plus seven 𝑛. Negative one times nine is negative nine. Negative one times seven 𝑛, negative seven 𝑛. Moving to our negative two 𝑚, multiply that by nine, and we get negative 18𝑚. Negative two 𝑚 times seven 𝑛 equals negative 14𝑚𝑛.
Still inside the brackets, we need to multiply seven 𝑛 times negative two 𝑚. That equals negative 14𝑚𝑛. And we’re dealing with subtraction. We know that subtracting a negative can be represented with addition. And now inside the brackets, we have negative 14𝑚𝑛 plus 14𝑚𝑛. Those cancel out. The remaining portion of our brackets is negative nine minus seven 𝑛 minus 18𝑚. And we need to multiply this by nine minus two 𝑘. Nine times negative nine equals negative 81. Nine times negative seven 𝑛 equals negative 63𝑛. Nine times negative 18𝑚 equals negative 162𝑚.
And with our negative two 𝑘, negative two 𝑘 times negative nine equals 18𝑘. Negative two 𝑘 times negative seven 𝑛 equals positive 14𝑛𝑘. And negative two 𝑘 times negative 18𝑚 equals positive 36𝑚𝑘. And this value is our first term. We need to follow the same process to find terms two and three. For term two, we’re dealing with negative negative two 𝑚, which we can write as positive two 𝑚. And we need to multiply two 𝑚 by the determinant of this two- by-two matrix.
That determinant is found by multiplying negative two 𝑘 times nine plus seven 𝑛 and then subtracting seven 𝑛 times negative two 𝑘. Inside the brackets, negative two 𝑘 times nine is negative 18𝑘. Negative two 𝑘 times seven 𝑛 equals negative 14𝑛𝑘. We’re subtracting seven 𝑛 times negative two 𝑘, negative 14𝑛𝑘. Again, we’re subtracting a negative. We can make that more clear by making an addition. What we see now is negative 14𝑛𝑘 plus 14𝑛𝑘 inside the brackets. Adding those together, we get zero. Now we need to multiply negative 18𝑘 by two 𝑚. We’re left with negative 36𝑚𝑘. This is our second term.
We have the same procedure one more time. We’ll put it here: seven 𝑛 times the determinant of this two-by-two matrix. We find that by multiplying negative two 𝑘 times negative two 𝑚 and then subtracting negative one minus two 𝑚 times negative two 𝑘. Moving inside the brackets, negative two 𝑘 times negative two 𝑚 equals four 𝑚𝑘. We are subtracting. We’ll distribute this negative two 𝑘 over the negative one minus two 𝑚. Negative two 𝑘 times negative one equals two 𝑘. Negative two 𝑘 times negative two 𝑚 equals positive four 𝑚𝑘.
Now we need to distribute this negative sign to both the two 𝑘 and the four 𝑚𝑘. And inside the brackets, we’ll say four 𝑚𝑘 minus two 𝑘 minus four 𝑚𝑘. The four 𝑚𝑘s cancel out, and we have seven 𝑛 times negative two 𝑘, multiplied together, equals negative 14𝑛𝑘, which is our third term. The next thing we need to do is add our first, second, and third terms together. I’m gonna clear off some space to make room to do that. So if you wanna pause the video to copy any of this down, you can do that now.
So here we go! Our first time plus our second term plus our third term. Do you notice anything here? When we’re adding, we have a positive 14𝑛𝑘 and a negative 14𝑛𝑘; these will cancel out — as well, positive 36𝑚𝑘 and negative 36𝑚𝑘 — leaving us with negative 81 minus 63𝑛 minus 162𝑚 and 18𝑘. We can simplify this a little bit further by noticing that all the constants are factors of nine. If we take out negative nine, divide negative 81 by negative nine, and you get nine, divide negative 63𝑛 by negative nine, and you get positive seven 𝑛, divide negative 162𝑚 by negative nine, and you get positive 18𝑚. 18𝑘 divided by negative nine equals negative two 𝑘. The simplified expression of this determinant, the simplified form of this determinant, is negative nine times nine plus seven 𝑛 plus 18𝑚 minus two 𝑘.