Question Video: Finding the Order of a Matrix Transpose Mathematics • First Year of Secondary School

Video Transcript

If 𝑋 is a matrix of order four by one, then what is the order of the matrix the transpose of 𝑋?

In this question, we are told that a matrix 𝑋 has order four by one. And we want to use this information to determine the order of the transpose of matrix 𝑋. To do this, let’s start by recalling what we mean by the order of a matrix. The first number tells us the number of rows the matrix has, and the second number is the number of columns. This means that matrix 𝑋 has four rows and one column. We can use this to write down that matrix 𝑋 is a four-by-one matrix with unknown entries 𝑎, 𝑏, 𝑐, and 𝑑.

We also need to recall that we can find the transpose of a matrix 𝑋 by rewriting each row of matrix 𝑋 as the corresponding column in the transpose matrix. In other words, we switch the rows and columns. We can use our representation of matrix 𝑋 to find a representation of the transpose of 𝑋. We will do this by writing each row of 𝑋 as the column in this new matrix called the transpose of 𝑋. The first row of 𝑋 only contains a single entry 𝑎. We write this as the first column of the transpose of 𝑋. We can follow the same process for the second row of 𝑋; its only entry is 𝑏, so we write this as the second column of the transpose of 𝑋.

We can follow this same process for the remaining rows of 𝑋. We find that the third column of the transpose of 𝑋 contains only 𝑐 and the fourth column contains only 𝑑. This gives us the following representation of the transpose of 𝑋. We can see that it has one row and four columns. Therefore, the transpose of 𝑋 is a one-by-four matrix.

It is worth noting that we do not need to directly represent the transpose of a matrix to answer questions of this form. Instead, we can note that taking a transpose switches the number of rows of a matrix with its number of columns. So, a matrix of order 𝑚 by 𝑛, when transposed, will become a matrix of order 𝑛 by 𝑚. We can then use this result to directly answer the question. 𝑋 is a four-by-one matrix, so when we take its transpose, we will switch the order around to obtain a one-by-four matrix.