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## Linear algebra

### Unit 2: Lesson 5

Finding inverses and determinants

# Determinants along other rows/cols

Finding the determinant by going along other rows or columns. Created by Sal Khan.

## Want to join the conversation?

• Is it possible to find the determinant by going down the diagonal?
• Yes, and no. One method of finding the determinant of an nXn matrix is to reduce it to row echelon form. It should be in triangular form with non-zeros on the main diagonal and zeros below the diagonal, such that it looks like:
[1 3 5 6]
[0 2 6 1]
[0 0 3 9]
[0 0 0 3] pretend those row vectors are combined to create a 4x4 matrix. Once it is in that form so that it appears like:
1 3 5 6
0 2 6 1
0 0 3 9
0 0 0 3
Then the determinant = the product of the entries along the diagonal, such that determinant = (1)(2)(3)(3) = 18.
Note* if the main diagonal contains a zero the determinant is also 0, thus the matrix is not invertible.
Hope that was clear enough to help.
• Am I the only one that doesn't see why it works going down any row or column, and why you need to switch signs?
`|a₁ a₂ a₃|`
`|b₁ b₂ b₃|`
`|c₁ c₂ c₃|`
If we multiply it out, we get:
`a₁b₂c₃ - a₁b₃c₂ + a₂b₃c₁ - a₂b₁c₃ + a₃b₁c₂ - a₃b₂c₁`
Notice how each term has an a, b, and c in it and also has a 1, 2, and 3 in it. This is why it works to use any row or column. Whichever row or column you use is the one you're factoring out. So, let's say we want to use the 2 column. In doing so, we factor out all of the 2's:
`a₂(b₃c₁-b₁c₃) + b₂(a₁c₃-a₃c₁) + c₂(a₃b₁-a₁b₃)`
Notice that in each of the parenthesis, we have the equation of a 2x2 determinant now. However, 2 of them go 31-13 while the other goes 13-31. If we want it to be the determinant of a sub-matrix, we need them to be in the order 13-31, so we get:
`-a₂(b₁c₃-b₃c₁) + b₂(a₁c₃-a₃c₁) - c₂(a₁b₃-a₃b₁)`
This is why it switches signs depending on which column or row you choose.
• Time stamp not showing for me, but at about 25% Sal give the formula Sign(i, j) = -1^(i+j). That should be written (-1)^(i+j) so the negative will also be raised to the power.
• If having 0s make finding the determinant easier, is it better to reduce it to rref first and then take the dets?
• Yes, but each row operation changes the determinant by some scalar factor, and you have to keep track of all of the factors.
• if the entries are variables instead of number should we use the same process??
• Yes. You treat the variable just as you would the mathematical object it represents.
• Can you find this on a graphing calculator?
(1 vote)
• Yes. You can do it on a graphic calculator...I have a Casio ClassPad. And all I have to do is type in the matrix and demand the determinant. However, exams and lecturers require us to be able to do these problems without a calculator...so I think its better to learn this method anyways
• So the row or column that you choose to perform the cofactor expansions will be your pivot row to perform row or column operations from? Or can you switch your pivot row or column at any time during the process? Will that effect the process of computing the determinant?
(1 vote)
• Once you pick a row (or column), you must find all the products that start with an element from that row (or column).
(1 vote)
• For this method, could we also use coloumn or row reduction to for example, get a row of [1 2 0 0] into [1 0 0 0] by doing C2 - 2C1 = C2 and then go about your method?
(1 vote)
• The determinant of a row reduced matrix must be the same (or at least both 0 or both non 0) as the one for the original, because either both A and rref(A) are invertible or neither is.
(1 vote)
• The videos in this section are beautiful. Excellent this one too, I didn't know you could keep changing where you did the determinant. I usually chose one row for the overall determinant and all the "inner" determinants on the same row.
(1 vote)
• Is this process also considered finding the M32 and C32 of a 4 x 4 matrix?