How to prove that only square matrix can have inverse matrix

How to prove that only square matrix can have inverse matrix

In fact, it should be: only square matrix can have inverse matrix
Because the definition of inverse matrix requires AB = Ba = I
If the identity matrix I is a square matrix, then according to the requirements of matrix multiplication, a and B can only be a square matrix
In fact, for non square matrix, we can define generalized inverse matrix

Find the inverse of the following matrix
1 2 3
2 2 1
3 4 3

1 2 3 1 0 02 2 1 0 1 03 4 3 0 0 1r2-2r1.r3-3r11 2 3 1 0 00 -2 -5 -2 1 00 -2 -6 -3 0 1r1+r2,r3-r21 0 -2 -1 1 00 -2 -5 -2 1 00 0 -1 -1 -1 1r1-2r3,r2-5r31 0 0 1 3 -20 -2 0 3 6 -50 0 -1 -1 -1 1(-1/2)r2,(-...

Judge whether the following matrix is the inverse matrix, if so, request the inverse matrix
1.[cosθ sinθ；-sinθ cosθ]
2.[cosθ sinθ；sinθ -cosθ]

cosθ sinθ
-sinθ cosθ
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First of all, he has an inverse matrix to judge whether his determinant is 0 or not
That is, (COS θ) ^ 2 + (sin θ) ^ 2 = 1 is not zero, so there is an inverse matrix
Second, the inverse matrix of the second-order matrix is the rule that the main diagonals are exchanged and the sub diagonals are negative
So the inverse matrix is
cosθ -sinθ
sinθ cosθ
The second is the same