Video: Finding the Order of a Matrix Using the Matrix Multiplication Property

Video Transcript

Given that 𝐴 is a matrix of order two by three and 𝐵 transpose is a matrix of order one by three, find the order of the matrix 𝐴𝐵, if possible.

First, we need to determine whether it’s actually possible to find the product of the two matrices 𝐴 and 𝐵. Two matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second. 𝐴 is a matrix of order two by three. This means that it has two rows and three columns. 𝐵 transpose is a matrix of order one by three, meaning it has one row and three columns. What about the matrix 𝐵? Well, the transpose of a matrix is found by swapping its rows and columns around. Therefore, the matrix 𝐵 will have three rows and one column.

In order to be able to find the product 𝐴𝐵, we need the number of columns of the first matrix 𝐴 to be equal to the number of rows of the second matrix 𝐵. They’re both equal to three. And therefore, it is possible to find the product 𝐴𝐵.

Now, we need to find the order of this matrix. If we multiply matrix of order 𝑚 by 𝑛 by matrix of order 𝑛 by 𝑝, then the resulting matrix will have the same number of rows as the first matrix 𝑚 and the same number of columns as the second matrix 𝑝. It will be of order 𝑚 by 𝑝.

The number of rows in the first matrix 𝐴 is two. The number of columns in the second matrix 𝐵 is one. Therefore, the order of the matrix 𝐴𝐵 is two by one.