A=[1 2
   3 4]
B=[7 2
   3 2]
@show(A .* B)    # Elementwise times is the "dot star"
@show(A * B);    # Matmul is just the "star"

A = [ 2  -1  5
      3   4  4
     -4  -2  0]
B = [ 1   0  -2
      1  -5   1
     -3   0   3]
C = A * B

B*A

A*B - B*A

A*(A^2 + 2*A + inv(A)*10) 

(A^2 + 2*A + inv(A)*10)  * A

A

A[2,3] # 2nd row, 3rd col

A[2,:] # 2nd row of A

B

B[:,1] # 1st column of B

dot(A[2,:], B[:,1])

A[2,:] ⋅ B[:,1]   # type \cdot + tab

α = A[2,3] # \alpha + tab

C[2,1]

A[2,:]' * B[:,1]  # yet another way to get a dot product

function matmul_ijk(A,B)
   m,n = size(A)
   n2,p = size(B)
   if n≠n2 error("No good, n=$n ≠ n2=$(n2)") end
   
   C = fill(0,m,p) # m x p "zeros" matrix
    
   for i=1:m, j=1:p, k=1:n
      C[i,j] += A[i,k]*B[k,j]   # shorthand for C[i,j] = C[i,j] + A[i,k]*B[k,j] 
   end
    
   return C  
end

matmul_ijk(A,B)

A*B

B[:,1]

A * B[:,1]

using Interact

@manipulate for j=1:3
    A * B[:,j]
end

[ A*B[:,1]  A*B[:,2]  A*B[:,3] ]

A*B

A

A * [ -1  0  0
       0  0  1
       0  1  0  ]

A * [ 0 1 0
      1 0 0
      0 0 1 ]

A

A[1,:]

A

A[1,:] * B

A[1,:]'

A[1,:]' * B

C

B * A[1,:]'

[ A[1,:]'*B 
  A[2,:]'*B
  A[3,:]'*B ] == C

B

[ 1 0 0
  0 1 0
  2 0 1 ] * B

[ 1 0 0
 -1 1 0
  3 0 1 ] * B

A[:,1] * B[1,:]'

A[:,1] * B[1,:]' + A[:,2] * B[2,:]' + A[:,3] * B[3,:]' == C