Question Video: Determining the Type of a Matrix Mathematics

Video Transcript

If the matrix 𝐴 is equal to one, zero, zero, one, which of the following is true? The matrix 𝐴 is a row matrix. The matrix 𝐴 is a column matrix. The matrix 𝐴 is an identity matrix. Or the matrix 𝐴 is a zero matrix.

Let’s consider each of these four statements and remind ourselves of the features of the type of matrix they refer to. We’ll start with the final statement: the matrix 𝐴 is a zero matrix. A zero matrix can have any dimensions, but every element within the matrix must be equal to zero. Looking at our matrix 𝐴, we can see that it does indeed have two elements which are equal to zero. But the other two elements, the elements which are on what we call the leading diagonal, are not equal to zero. So matrix 𝐴 is not a zero matrix because it doesn’t have every element equal to zero.

Let’s now consider the top statement: the matrix 𝐴 is a row matrix. A row matrix or row vector is a matrix of order one by 𝑛. It’s simply a matrix which has only one row, although it can have any number of columns. In its general form, it would look something like this, and the elements would be labeled as 𝑎 one one, 𝑎 one two, all the way up to 𝑎 one 𝑛. Looking at our matrix 𝐴, we can see that it has not one but two rows. And therefore, the matrix 𝐴 is not a row matrix.

We’ll now consider the second statement: the matrix 𝐴 is a column matrix. Well, a column matrix or a column vector is a matrix of order 𝑚 by one. So it can have any number of rows, but it must have only one column. In its general form, it would look a little something like this. And we could label the elements as 𝑎 one one, 𝑎 two one, all the way down to 𝑎 𝑚 one. Looking at our matrix 𝐴, it’s easy to see that this matrix has two columns, not one. And therefore, the matrix 𝐴 is not a column matrix.

We’re left with only one possibility. Is the matrix 𝐴 an identity matrix? Well, an identity matrix is, first of all, a diagonal matrix which also means it must be square. Being a diagonal matrix means that all of the elements not on the leading diagonal, that’s the diagonal from the top left to the bottom right, must be equal to zero. We can see that this is the case for our matrix 𝐴. Furthermore, in order to be an identity matrix, rather than just a diagonal matrix, all of the elements on the leading diagonal must be equal to one. Looking at our matrix 𝐴, we can see that this is true for the two elements on the leading diagonal.

So we have a diagonal matrix. All of the elements not on the leading diagonal are equal to zero. And all of the elements that are on the leading diagonal are equal to one, which means that the matrix 𝐴 is an identity matrix.

In fact, because it’s a matrix of order two by two, we sometimes refer to it as 𝐼 two, the two-by-two identity matrix. The only statement which is true then is this one: the matrix 𝐴 is an identity matrix.