Well, I'm curently learning Linear Algebra from MIT OpenCourse 18.06 with Profesor Strang.
I'm currently at video 20 about Cramer's rule, about the equation for inverse of a matrix and I quite don't understand the 'proof'.
the proof starts about 3:30
Well, I quite don't get one thing:
We start from formula $A^{-1} = \frac{1}{detA} C^{T}$ where $C$ is matrix of cofactors of matrix A.
We change it to : $A C^T = (detA)I$
$$ \pmatrix{ a_{11} & \dots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nn} } \pmatrix{ C_{11} & \dots & C_{n1} \\ \vdots & \ddots & \vdots \\ C_{1n} & \dots & C_{nn} } = \pmatrix{ detA & \dots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \dots & detA } $$
I really just can't understand how the last matrix should be the same as $ (detA) I$
shouldnt it be $ (detA)^ {n} I$ because we have $detA$ on every line of identity matrix?
Also can't understand why there are zeros on non diagonal ...
Is there something I missed??
EDIT: I already understood that about getting detA before matrix.. Such a basic thing.. But still i'm not able to understand the second part...
So my question now is:
Why when we multiply row a from A and row b from C when $ a \neq b$ why it is 0 ??