Why The Statement The Second Row Of Ab Is The Second Row Of A Multiplied By B Is Not Always True In Matrix Product – An Explanation.

The second row of AB is the second row of A multiplied on the right by B.

True because upon multiplying out AB, you will see that the second row of AB is the second row of A multiplied on the right by B.

The statement The second row of AB is the second row of A multiplied on the right by B is not true in general.

To see why, let A and B be two matrices of compatible sizes such that A is an m x n matrix and B is an n x p matrix. Then, the matrix product AB is an m x p matrix, and its entries are given by:

(AB){i,j} = \sum{k=1}^n a_{i,k} b_{k,j}

For any given i, the ith row of AB is a linear combination of the rows of B with coefficients given by the entries of the ith row of A. In other words, the ith row of AB is the ith row of A multiplied on the right by B.

However, the statement The second row of AB is the second row of A multiplied on the right by B is not necessarily true, since it specifies the second row of AB, rather than the ith row for some arbitrary i.

For example, consider the matrices:

A = [1 0]
[0 1]

B = [0 1]
[1 0]

Then, AB = [0 1]
[1 0]

The second row of AB is [1 0], but the second row of A multiplied on the right by B is [0 1]. So, the statement is not true in general.

However, it is true that each row of AB is a linear combination of the rows of B with coefficients given by the entries of the corresponding row of A. In other words, each row of AB is the corresponding row of A multiplied on the right by B.

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