If A and B are 3 x 3 matrices and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3]
False because that expression represents the sum of those three vectors which results in one vector having dimensions 3 x 1. The product AB is a 3 x 3 matrix so from this immediately, we can say that the statement is not true.
The statement If A and B are 3 x 3 matrices and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3] is not true in general.
In fact, for the matrix product AB to be defined, the number of columns of A must equal the number of rows of B. Since B is a 3 x 3 matrix with three columns, A must have three rows. So, let’s assume that A is a 3 x 3 matrix, as required.
Then, the matrix product AB is a 3 x 3 matrix whose entries are given by:
(AB){i,j} = \sum{k=1}^3 a_{i,k} b_{k,j}
Using the fact that B = [b1 b2 b3], we can write this as:
(AB){i,j} = \sum{k=1}^3 a_{i,k} [b1 b2 b3]_{k,j}
= a_{i,1} b_{1,j} + a_{i,2} b_{2,j} + a_{i,3} b_{3,j}
= [Ab1]{i,j} + [Ab2]{i,j} + [Ab3]_{i,j}
So, we have shown that:
(AB){i,j} = [Ab1]{i,j} + [Ab2]{i,j} + [Ab3]{i,j}
for all i,j, which means that the matrices AB and [Ab1 + Ab2 + Ab3] have the same entries. However, this does not necessarily mean that they are equal as matrices, since two matrices are equal if and only if they have the same entries in the same positions.
Therefore, the statement If A and B are 3 x 3 matrices and B = [b1 b2 b3], then AB = [Ab1 + Ab2 + Ab3] is not true in general, but we have shown that the two matrices have the same entries.
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