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### Course: Linear algebra>Unit 2

Lesson 6: More determinant depth

# Simpler 4x4 determinant

Calculating a 4x4 determinant by putting in in upper triangular form first. Created by Sal Khan.

## Want to join the conversation?

• Hello Sal. I know I'm wrong and the answer is probably staring me in the face! Where's the fallacy in my thinking: As I understand it, a square matrix whose determinant is not zero is invertible. Therefore, using row operations, it can be reduced to having all its column vectors as pivot vectors. That's equvialent to an upper triangular matrix, with the main diagonal elements equal to 1. If normal row operations do not change the determinant, the determinant will be -1. HELP! :)
• Sal was a bit unfortunate when he said that "row operations do not change the determinant". The proof he gave was that row operations of the form [Ri -c*Rj -> Ri] (that is, replacing row i with row i minus row j times a scalar c) do not change the determinant.

But there are row operations of different kind, such as
k*Ri -c*Rj -> Ri (That is, replacing row i with row i times a scalar k minus row j times a scalar c).

What can be proved is that operations of this kind do change the determinant. In fact, they multiply the determinant by k.

And when you put an invertible matrix in RREF (that is, you turn it into an identity matrix), you must do these kinds of operations that scale the determinant. And they always end up scaling the determinant back to one, which is the determinant of any identity matrix.
• If i do two swaps, does the second essentially cancel out the first and thus the determinant becomes positive again?
• Yes, doing two swaps one after the other, will not change the determinant.
• Can i use this method for any square matrices? thanks
• Yes. Since the determinant remains constant when you subtract a scaled row from another row, this can be used for any square matrix.
• Det A =5. B=4A.then find Det B=?
• It depends on the size of A and B. Multiplying a matrix by a scalar, is the same as multiplying every row of that matrix by that scalar, and note, that multiplying a single row by a scalar is equivalent to multiplying the determinant of the original matrix by that scalar. For example, the determinant of
[1 0] = 1, the determinant of [2 0] = 2, and the determinant of [2 0] = 4.
[0 1] [0 1] [0 2]
Notice with each step, I multiplied a single row by 2, and the determinant increased by a factor of that scalar 2. And the last step, I multiplied all of the rows by 2, which had the same result as multiplying the entire matrix by 2.
Long story short, multiplying by a scalar on an entire matrix, multiplies each row by that scalar, so the more rows it has (or the bigger the size of the square matrix), the more times you are multiplying by that scalar.
Example, if A is 3x3, and Det(A) = 5, B=2A, then Det(B) = 2^3*5=40.
The formula for Det(kA), where k is a scalar, and A is an nxn matrix is as follows:
Det(kA)=k^n*Det(A).
• can someone please tell me where is the proof of determinant when swapping rows? I can't find it:(
• The above link does not contain the proof for swapping of matrix rows
• How do I find a determinant of a 4x4 matrix that has an unknown "k" in the last row?
Its not possible to do it by the upper triangular method as that would take away the "k" i'm trying to find. Help!
(1 vote)
• In the previous video Sal said that you could make it lower triangular and that would also work.
• Can I swap any row and the determinant will be negative? Or just the mid ones?
• You can swap any two rows and it will have the effect of negating the determinant.
• At he swaps a row with another and add a -ve sign. If further down the computation process, if we have to swap another row, do we add another "-" sign thus making the Det. positive again..or does the minus stay no matter the no. of row-swaps?

Thanks
• I also have this same question! Is anybody able to clear this up for us? Sal states that switching two rows (once) means we must multiply the resulting determinant by -1 to get the determinant of the original matrix. What if we switch rows more than once? Do we have to multiply the resulting determinant by -1 for each row swap?
(1 vote)
• If i have found the Det of the Matrix Det A how do you find the Det A^3
(1 vote)
• det(A^3)=det(AAA)=det(A)*det(A)*det(A)=[det(A)]^3.
In other words, cube the number that you found from det(A).