If you have worked steadily through the Programme, you should have no difficulty with this exerc

If you have worked steadily through the Programme, you...

Key Concepts

Determinant

A determinant is a scalar value that is computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible, and it plays a crucial role in solving systems of linear equations, evaluating eigenvalues, and understanding linear transformations.

3x3 Matrix Determinant Calculation

For a 3x3 matrix, the determinant can be calculated using systematic methods such as cofactor expansion (also known as Laplace's expansion) or the Rule of Sarrus. These methods organize the computation into sums and differences of products of matrix elements, ensuring a clear path to the result.

Rule of Sarrus

The Rule of Sarrus is a specific shortcut technique used for evaluating the determinant of a 3x3 matrix. It involves extending the matrix by rewriting its first two columns and then summing the products of the diagonals from the top left to bottom right, while subtracting the products of the diagonals from the bottom left to top right.

Transcript

00:01 So in this problem, we're given three different four by four matrices and asked to calculate the determinant of each one.

00:15 So the first trick here is to find a row or column that's not too bad to work with, has some zeros in it or something that you can kind of reduce or minimize the number of calculations.

00:26 And i don't really have that on any of these.

00:29 So i'm just going to use row two here.

00:35 And remember, this is plus, this is minus.

00:38 This is plus, this is minus, this is plus.

00:41 So, the determinant is going to be minus two times the determinant of the three by three that's remaining after i take out that row in that column.

00:53 Minus one, three, minus four, three, minus five, four, two, minus one, four, plus one times the determinant of, take that row and that column out.

01:10 So that's 2.

01:13 3 minus 4.

01:17 3 .3 minus 5.

01:28 Got lost there, didn't i, for a second.

01:33 3 minus 5, 4, 5, 5, 4.

01:37 5, minus 1, 4.

01:40 Okay.

01:42 Then i have minus a negative 2 times the determinant of 2 minus 1 minus 4, 3, 3, 3, 4, 5, 2.

01:57 Two, four.

01:59 And lastly, plus one times the determinant of, take out that row in that last column, two, minus one, three, three, three, minus five, five, two minus one.

02:19 Okay, so that means i now have minus two times, let's see.

02:27 Something in this first matrix that i can use very easily, well i'm just going to use this column right here remember that's plus minus that's plus minus plus right and so i have minus four times the determinant of the two by two that's left three minus five two minus one minus four times the determinant of minus one three two minus one and then minus four times the determinant of minus one, three, three, minus five.

03:16 Okay.

03:17 This next matrix.

03:18 Well, let's see.

03:20 Not really any really good one easy to use.

03:23 I'll just use this top row.

03:25 Remember, this is plus, minus, plus.

03:28 So, i have plus.

03:31 One times anything is itself, right? so that's two times the determinant of minus five, four, minus one, four, minus one, four, minus three times the determinant of three, four, five, four, and then minus four times the determinant of three minus five, five, minus one.

04:08 Okay.

04:10 Let's see over here, i'll tell you what, i'll use this last column again, kind of like i did on the first one.

04:16 So minus a minus two, so that's plus 2 times minus 4 times the determinant of 3352 minus 4 times the determinant of 2 minus 4 times the determinant of 2 minus 1 5 2 plus okay and then i on the last one over here, i'm just going to use this top row, because nothing really a whole lot better to use.

05:10 Okay.

05:12 I have plus one times it, so i just have plus two times the determinant of what's left, three minus five, two, minus one, minus a minus one, times the determinant of three minus five, 5 minus 1 plus 3 times the determinant of 3, 352.

05:45 Okay, so this gives me what? well, minus 2 times minus 4, that's 8 times.

05:54 Remember the determinant on a 2 by 2, we go this way, and then we go minus that way, right, on the two diagonals.

06:02 So that's minus 3, minus a negative 10, so that's plus 10.

06:09 And then minus 2 times 4, so that's plus 8 times minus 1 is 1, minus 6.

06:23 And minus 2 times minus 4, this is plus.

06:30 This is plus.

06:33 That's minus 8 times minus 1 times 5 is 5.

06:40 3 times 3 is 9.

06:42 And then i got plus two times minus five times four that's minus 20 plus four because minus a negative four okay minus three times 12 minus 20 minus four times minus three times minus three plus 25 then i have minus eight times 2 times minus 4 is 8, 6 minus 15, and then minus 8 again times 4 minus 5, 4 plus 5, sorry, and then plus 8 times 6 plus 3, and then for the last row i got plus 2 times minus 3 plus 10, plus minus and minus 1, that's plus 1.

08:06 That's just the determinant minus 3 plus 25 and then plus 3 times 6 minus 15.

08:18 Put all this together and you're going to get 33 for the answer without determinant.

08:27 Okay.

08:29 Now, let's look at the next matrix.

08:33 So let's see.

08:34 I got a zero here i could work with and a bunch of ones and twos in that second row.

08:39 So let's use that.

08:42 Remember, that's plus.

08:43 So this is minus.

08:44 This is plus, minus, plus.

08:46 All right.

08:49 So here we go.

08:50 Minus and minus one.

08:51 That's plus one times the determinant of, this was minus a minus one.

09:02 Determinant of minus one, four, minus three, two, three minus one, minus two, two, minus three, plus one times of the determinant of, take out that row in that second column, 2, 4, minus 3, 3, 3 minus 1, 1, 2 minus 3.

09:32 And then lastly, it will be plus 2 times the determinant of 2 minus 1, 4, 3, 2, 3, 3, 3, 1, 2, 2...

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