A Determinant Trick #math #linearalgebra #determinant

Asking for the determinant of a huge matrix like this seems like a very difficult computation or at least it feels like it's very cumbersome. But it turns out there's a really nice quick way to do this using an identity. So the first thing you notice is you can write this matrix as the identity matrix which has ones on the diagonal plus a matrix that contains all ones everywhere. And to abbreviate that we can write that as the identity matrix of dimension 7 plus this vector right here transpose times the vector itself. And this vector contains exactly seven ones in it. So when you multiply these two, this being a 7x1 matrix and this being a 1x7 matrix, you get a 7x7 matrix of all ones. Okay. So the really cool property which I actually mentioned in a previous video is that if you take the determinant of the identity matrix times the product of two matrices, you can actually switch the order of these and add the identity matrix of the appropriate size and compute the determinant of that instead. Now since this is a 7x1 matrix and this is a 1x7 matrix when we do the swap we get a 1x 7 matrix * a 7x1 matrix and the result is therefore a 1x1 matrix or a scalar number. So the dimension of the identity matrix of the appropriate size is actually a dimension one matrix. So it's just the number one and we're left with computing this dotproduct and since there's exactly seven 1's in each matrix the dot productduct is seven. And so we get 1 + 7 which is eight.