If 𝐴 is a two-by-three matrix and 𝐵 is a three-by-two matrix, can you add the matrices 𝐴 and 𝐵?
We recall first that in order to add two matrices, they must be of the same order. This means they need to have the same number of rows and the same number of columns as one another. In general, the order of a matrix is often written as 𝑛 by 𝑚, where 𝑛 represents the number of rows in the matrix and 𝑚 represents the number of columns.
We’re given in the question the orders of matrices 𝐴 and 𝐵. 𝐴 is a two-by-three matrix, so it has two rows and three columns, whereas 𝐵 is a three-by-two matrix, so it has three rows and two columns. The matrices are therefore not of the same order, and so it is not possible to add them.
The reason we can’t add matrices of different orders is because the way we do add matrices is by adding corresponding elements in the same position. So, for example, we add the elements in the first row and first column to find the element in the first row and first column of the matrix representing their sum. If matrices have different orders, then there will be some elements without corresponding elements in the other matrix.
So our answer to the question “can you add the matrices 𝐴 and 𝐵?” is no.