Question

How to derive Cramer's Rule?

Answer 1

I am supposed to use these three pieces of information:

1. You can find the inverse of a matrix by augmenting the matrix to the identity matrix, and using Gaussian Elimination to row-reduce to (I | A^-1)
2. A^-1 = 1/det(A) * adj(A)
3. Laplace's formula for determinant

I understand these concepts separately but I don't know how to put them all together :/

Answer 2

My attempt so far:

Ax = b, therefore x = A^-1 * b

Answer 3

Using adjoint rule,

x = A^-1 * 1/det(A) * adj(A) * b

Answer 4

Replace i-th column with x to obtain A_i matrix

Answer 5

Label that column as x_i

Answer 6

I think I use laplace formula using that column but I'm not sure how to write it out

Answer 7

@TheSmartOne @Directrix

Answer 8

i think it might be in here https://youtu.be/QNpj-gOXW9M

or you can ask @Empty @Kainui @ganeshie8