Can we multiply a 2x3 matrix by a 2x2 matrix?

False. The matrices must conform. You can multiply a m x n matrix by a n x k matrix.

The output is m x k. The inner dimensions must match. The result is the outer dimensions.

2x2 by 2x3 yes a 2x3 2x3 by 2x2 no. Obviously true if m x n by n x m or two square matrices. Note that the shapes of the two resulting matrices will differ unless both are square.

False A could be a 4 x 2 matrix and B could be a 2 x 7 matrix so AB would give a 4 x 7 matrix result but BA would be meaningless.

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