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.
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