### Video Transcript

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

In this question, we are told that
a matrix π has order four by one. And we want to use this information
to determine the order of the transpose of matrix π. To do this, letβs 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 π has four
rows and one column. We can use this to write down that
matrix π is a four-by-one matrix with unknown entries π, π, π, and π.

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

We can follow this same process for
the remaining rows of π. We find that the third column of
the transpose of π contains only π and the fourth column contains only π. This gives us the following
representation of the transpose of π. We can see that it has one row and
four columns. Therefore, the transpose of π 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 π by π,
when transposed, will become a matrix of order π by π. We can then use this result to
directly answer the question. π 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.