Matrix Multiplication: Debunking The False Claim That Ab = [A1B1 A2B2] For 2 X 2 Matrices A And B.

If A and B are 2 x 2 with columns a1, a2 and b1, b2, respectively, then AB = [a1b1 a2b2].

False because it violates the definition of the matrix AB

The statement If A and B are 2 x 2 with columns a1, a2 and b1, b2, respectively, then AB = [a1b1 a2b2] is False.

The product of two matrices A and B is defined as follows: if A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix whose entries are given by:

(AB)ij = Σk=1 to n (AikBkj)

In other words, to compute the entry in the ith row and jth column of AB, we take the dot product of the ith row of A with the jth column of B.

In the case of 2 x 2 matrices A and B with columns a1, a2 and b1, b2, respectively, we have:

A = [a1 a2], where a1 and a2 are column vectors of length 2, and

B = [b1 b2], where b1 and b2 are column vectors of length 2.

Therefore, the product AB is given by:

AB = [a1 a2][b1 b2] = [a1b1 + a2b2 a1b2 + a2b2],

which is a 2 x 2 matrix, not a 1 x 2 matrix as claimed in the statement. So the statement is False.

More Answers:
Matrix Transpose: Proving A^T + B^T = (A + B)^T.
Mastering Matrix Mathematics: The Distributive Property Of Matrix Multiplication Over Addition.
The True Statement About Matrix Multiplication: Columns Of Ab As Linear Combinations Of Columns Of B Using Weights From Corresponding Columns Of A.

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