Which of the following matrices must be a square matrix? (A) Identity matrix, (B) diagonal matrix, (C) upper triangular matrix, (D) row matrix, (E) zero matrix. And the choices are only (D) and (E); (A), (B), and (C); (A), (D), and (E); only (A) and (B); or all the choices must be square matrices.
We know that a matrix is an array of numbers, but what is a square matrix? Well, for a matrix with 𝑚 rows and 𝑛 columns, we say that the matrix is a square matrix if 𝑚 is equal to 𝑛. In other words, the matrix has the same number of rows as it does columns. An example of a square matrix is two, negative one, four, zero. This matrix has two rows and two columns, whereas the matrix seven, five, one, negative six, zero, two is not a square matrix because it doesn’t have the same number of rows as columns. It has two rows and three columns, so this is not a square matrix. So now we’ve recalled what a square matrix is, let’s have a look at these types of matrices to decide which of these are going to be square matrices.
Let’s begin by thinking about the identity matrix. For any integer 𝑛, there is an identity matrix of order 𝑛 by 𝑛. Remember that the order of the matrix just means the size of the matrix. So an 𝑛-by-𝑛 matrix has 𝑛 rows and 𝑛 columns. The identity matrix has all the elements as zero except for the elements in the main diagonal, which are all one. For example, the three-by-three identity matrix is one, zero, zero, zero, one, zero, zero, zero, one. So because by definition the identity matrix has order 𝑛 by 𝑛, it’s always going to have the same number of rows as it does columns. And therefore, the identity matrix must be a square matrix.
So now let’s consider a diagonal matrix. In a diagonal matrix, all of the entries outside of the main diagonal are zero. This is different to the identity matrix because these elements on the diagonal could be anything, whereas in the identity matrix all the elements on the diagonal had to be one. An example of a diagonal matrix is the matrix five, zero, zero, zero, negative one, zero, zero, zero, nine. Now, a matrix can only have a main diagonal from top left to bottom right when it’s a square matrix. Therefore, a diagonal matrix must be a square matrix.
Now let’s consider an upper triangular matrix. An upper triangular matrix is a matrix whose entries below the main diagonal are zero. An example of an upper triangular matrix would be the matrix four, three, six, seven, zero, one, four, negative two, zero, zero, one, three, zero, zero, zero, six. You can see that below the main diagonal from top left to bottom right, all of the entries are zero. So if a matrix is an upper triangular matrix, it must have to have a main diagonal. And because it has a main diagonal, it therefore must be a square matrix.
So now let’s consider whether a row matrix must be a square matrix. A row matrix is a matrix where the elements are all arranged in one row such as the matrix two, zero, one, four, five. So for a matrix with 𝑚 rows and 𝑛 columns, for a row matrix, we must have 𝑚 equal to one. That means the matrix has one row. So as we can see just by this example, a row matrix is not necessarily going to be a square matrix. Having said that, if we consider the row matrix that contains only a single element such as nine, this is a row matrix and also a square matrix because it has one row and one column. But we can’t say that all row matrices are going to be square matrices.
So finally, let’s consider whether the zero matrix must be a square matrix. In a zero matrix, all the entries in the matrix are zero. And a zero matrix can be of any size as long as all the entries are zero. So the matrix zero, zero, zero, zero is a zero matrix, but so is the matrix zero, zero, zero, zero, zero, zero. So, as we can see just by these examples, a zero matrix is not necessarily going to be a square matrix.
So we can see that the matrices which must be square matrices are an identity matrix, a diagonal matrix, and an upper triangular matrix. Therefore, the answer is (A), (B), and (C).