Find the area of the triangle 𝐴𝐵𝐶 with vertices 𝐴: one, four; 𝐵: negative four, five; and 𝐶: negative four, negative five.
At this problem, we can actually see that we can actually use the determinant to help us find the area of the triangle 𝐴𝐵𝐶. And in order to do that, we’re actually gonna use an equation. So we can say that the area of a triangle with coordinates 𝑎, 𝑏; 𝑐, 𝑑; 𝑒, 𝑓 is the absolute value of 𝐴 given that 𝐴 is equal to a half of the determinant of the matrix 𝑎, 𝑏, one; 𝑐, 𝑑, one; 𝑒, 𝑓, one. So great, now that we have a formula that we can use, we can actually put our values in and calculate 𝐴.
What I’ve done first of all is I’ve actually labelled our coordinates. And this is gonna help us cause it can help us see what’s gonna go into our formula. So we’ve got 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓. So we can say that 𝐴 is equal to a half multiplied the determinant of the matrix. And the top row is gonna be one, four, one because that’s our 𝑎, 𝑏 and then one. Our next row will be negative four, five, one. Again, cause now we’re moving on to 𝑐, 𝑑. And then finally, our bottom row’s going to be negative four, negative five, one. And again now, we’ve substituted in our values for 𝑒, 𝑓.
Excellent! So now let’s calculate what the value of 𝐴 is. Before we can calculate what the value of the determinant is, we just remind ourselves that when we’re calculating the value of the determinant, we need the positive, negative, positive above our columns. And the reason that is cause it actually helps us decide whether our coefficient, so the top row number, is going to be positive or negative. Okay, great. So let’s get on with it.
So first of all, we know that our coefficient is gonna be one because that’s our first term in the first row. Okay, now what we do is we actually eliminate the same row and column that one’s in. And then we’re concerned with the submatrix of the bottom four numbers here, which I’ve put an orange box around. And we can just remind ourself that if we want to find the determinant of a-a submatrix, a two-by-two submatrix, that’s actually gonna be equal to 𝑎𝑑 minus 𝑏𝑐. Okay, great. So we can use that and we can work out ours. So we’re gonna have five multiplied by one, cause that’s like our 𝑎 and our 𝑑. And that’s minus one multiplied by negative five. Okay, great. We’ve done that part. Let me move on to the next part.
So our next coefficient is gonna be negative four. And that’s a four because it’s the second term in the first row. And it’s negative cause we know that the second column is going to be negative. So we’ve got negative four. So now we’re gonna multiply that negative four by the determinant of the submatrix negative four, one, negative four, one. And it’s that because we’ve actually deleted the values in the row that the four’s in and the column that the four’s in. So we’re gonna have negative four multiplied by one minus one multiplied by negative four. Okay, great.
And now we can move on to the final part where we’re gonna have positive one as a coefficient. And then inside, we’ve got negative four multiplied by negative five. That’s cause that’s like our 𝑎 and our 𝑑 in our submatrix. And that’s minus five multiplied by negative four. Okay, great. And then finally, we’re gonna divide the whole thing by two because if we look back to the original equation, it was a half multiplied by the determinant of our matrix.
Okay, great. So now let’s work out our values. Okay, so this gives us that 𝐴 equals one multiplied by 10 minus four multiplied by zero plus one multiplied by 40, all divided by two. So therefore, 𝐴 is equal to 50 over two which is equal to 25.
So now, we just check back cause it says the area of a triangle with the coordinates 𝑎, 𝑏; 𝑐, 𝑑; 𝑒, 𝑓 is the absolute value of 𝐴. Well, we look down to our value of 𝐴. It’s already positive 25. So we don’t have to worry about it cause in the absolute value we’re only looking for the nonnegative, or positive answers. So therefore, we can say that the area of triangle 𝐴𝐵𝐶 is equal to 25 square units.