Provide an appropriate response.
Suppose that A and B are two matrices such that A + B, A - B, and AB all exist. What can you conclude about the dimensions of A and B?
A) A is a row matrix and B is a column matrix.
B) A and B have the same dimension, but are not necessarily square matrices.
C) A and B are square matrices of the same dimension.
D) There are no dimensions of A and B that would make this possible.
To be able to add or subtract two matrices, they must be of the same dimension m x n
To be able to multiply matrix A by matrix B and obtain AB, the number of columns of A must equal the number of rows of B
Of the given choices, which one would satisfy both of these conditions?
The correct answer is B) A and B have the same dimension, but are not necessarily square matrices.
The correct choice is B) A and B have the same dimension, but are not necessarily square matrices.
To understand why, let's break down the statement. The question states that A + B exists, A - B exists, and AB exists.
For A + B to exist, the dimensions of A and B must be the same. In matrix addition, you can only add two matrices if they have the same number of rows and columns.
Similarly, for A - B to exist, the dimensions of A and B must be the same. In matrix subtraction, you can only subtract two matrices if they have the same number of rows and columns.
Now, for AB to exist, the number of columns in matrix A must be equal to the number of rows in matrix B. Matrix multiplication is only defined when the number of columns in the first matrix matches the number of rows in the second matrix.
From these conditions, we can conclude that A and B must have the same dimensions (same number of rows and columns) in order for all three operations (addition, subtraction, and multiplication) to be possible. However, they are not necessarily required to be square matrices (matrices with the same number of rows and columns).
Therefore, the correct answer is B) A and B have the same dimension, but are not necessarily square matrices.