How to Find the Determinant of a 5x5 Matrix

In this video I will teach you a shortcut method for finding the determinant of a 5x5 matrix using row operations, similar matrices and the properties of triangular matrices. This method is very easy to understand and with a little knowledge of Gaussian elimination and the properties of determinants it should prove easy to implement. I find this method much simpler than using cofactors.

[@ericl3347]

Thank you for uploading this and being thorough in your explanation! This method is by far the best and easiest to remember when trying to find the det(A). Thank you so much for this wonderful explanation! Please keep up the wonderful work and do more stuff like this for other topics!

[@olorunfemiomonigbehin389]

thank you for putting this up. You have helped a generation.

[@TheCompleteGuide1]

My pleasure!

[@gede1013]

thank u , people like u make the life better

[@aniccin]

This was like the easiest tutorial I've seen

[@NeronAzemi]

Thank you soo much. This really helped me loads

[@dimaignatiev6370]

Brilliant!thank you!

[@EatSpicySweet]

Thank u so much sir may God bless you

[@Stamnessj]

Hey! Do you have any tips when the matrix doesn't have such easy numbers? I had this one task where I had to multiply R2 with stuff like (-1/3 R3). This amounted in a matrix with entries like "-9/15", "5/3" etc. Although I got the right answer, the Gaussian elimination was hairy af, and it was hard noticing what row operations I needed to do.

[@Afterburn7]

No way to avoid it saddly.

[@ayo94563]

thank you!!!!!!!!

[@shaggyy6094]

Can I swap row then column to find the determinant?

[@mustafahameed7870]

Do the rules apply for column operations and swaps

[@jakobskovsndergard792]

if you multiply a row , shouldnt that affect your determinant? like if I for some reason needed to multiply row 3 by 1/2

[@kylitrixgames4980]

Yes, if you multiply a row with factor x, you need to devide the determinant by x.

[@arinzeanthony7447]

​@@kylitrixgames4980that will affect the whole matrix, we're talking about a row.

[@kirubaaharansaaihishan8132]

is this ok for 6x6 and 7x7 and so on....

[@user-xl1ig1bn6i]

Can you shiw us how to make the conjugate matrix of the first matrix (matrix A) and it's inverse matrix A^-1 using this trick, for a matrix of order 4x4 or 5x5?

[@muhammedafnan4741]

Thanks a lot

[@trustmeiamanengineer2075]

Thank u sir

[@user-wt7fu3qw4k]

I have same but 4 intead of 1 in the 5th row i use another method but the video is very good thx

[@rymichael7203]

If you do two swaps in the same move does the sign stay the same or change?

[@naulyd]

stays the same

[@rymichael7203]

@@naulyd Thanks

[@BruceWayne-dr9zh]

If we have to multiply any row by a constant and then add or subtract it from another row, do we also have to multiply the determinant by the constant??

[@eliasnino2337]

I think if you multiply a row by a constant, you have to divide your determinant by that constant at the end

[@vetianos]

could this be applied for det(A-λI) to find the characteristic polynomial?

[@kylitrixgames4980]

Don't do this for a 3x3 or less buddy. But yes you could

[@s888r]

You can use this method to find the determinant of a matrix of any order!

[@uwadiegwuemmanuel8809]

Thanks

[@arinzeanthony7447]

Isn't row and coloum operations like that supposed to affect the value of the determinat?

[@vijaysinghchauhan7079]

Can any matrix of any nxn order turn into a triangular matrix?

[@user-xl1ig1bn6i]

Yes, any nxn matrix can be reduced to an upper triangular matrix. You just need to choose your operations diligently.

[@uzomannaemekateflondon1144]

Love u sir

[@jxidospxzllxxljxixdpxlxlxk5523]

Wonderfull😍👍

[@deepseaofknowledge3482]

Can we apply it for lower triangular matrices?

[@sayanalal2999]

Can we?

[@sayanalal2999]

Determinant of transpose is equal to the determinant of matrix right?

[@BruceWayne-dr9zh]

For any triangular and diagonal matrix, det is the multiplication of the diagonal elements

[@zoterodisuke4398]

why is it negative though?

[@samratoli3111]

I like it but didn't understand

[@mumujibirb]

reminds me of REF

[@0nlinekjng922]

Uhm.

[@niall_al3059]

your method doesn't work completely. I only swapped rows once, so determinant went from + to - and yet the correct value for the determinant was a positive in the end. The multiplying the diagonal was fine but the sign was incorrect.

[@abhisheksingh-qp9yu]

are you mike boyd?

[@TheCompleteGuide1]

It is not widely known but all people with Scottish accents are secretly the same person

[@raymondvictoriofabian5443]

Very helpful video, but shouldn't (3)(-1)(-2)(3)(1) give you +18?

[@mathbyprofc8791]

Indeed. But don't forget about the negative sign that step 2 of the row reduction generated. So det(A) = -(3)(-1)(-2)(3)(1) or -18. I think of it like this. Every time you swap rows, it cost 1 negative.

[@raymondvictoriofabian5443]

@@mathbyprofc8791 Thank you for responding. I thought about it and saw how I was confused about determinants (overlooked that determinants change signs when rows are swapped).