Consider the following matrix 𝐴, which is equal to the three-by-three matrix one, zero, zero, four, zero, zero, one, zero, three. Which of the following is the type of matrix 𝐴? Is it option (A) identity matrix, option (B) row matrix, option (C) diagonal matrix, option (D) lower triangular matrix? Or is it option (E) upper triangular matrix?
In this question, we’re given a matrix 𝐴 and we need to determine which of five types of matrices is the matrix 𝐴. We’ll do this by going through each of the five types of matrices we’ve been given. Let’s start with option (A), which is an identity matrix. First, when we say an identity matrix, we mean an identity matrix for multiplication. And another way of saying this is 𝐼 is an identity matrix if 𝐼 is a square matrix and for any matrix 𝐵, which is also a square matrix, 𝐵 times 𝐼 equals 𝐼 times 𝐵 is equal to 𝐵. For any matrix 𝐵 of the same order as 𝐼, when we multiply on the right or on the left by 𝐼, we end up with 𝐵. 𝐼 is an identity matrix of multiplication.
And because of these properties, identity matrices all have the same form. First, they must be square matrices, which means they have the same number of rows and columns. And we can see this is true for option (A). Option (A) has three rows, and it has three columns. Next, every entry on the main diagonal of an identity matrix must be equal to one. These are the entries where the row number is equal to the column number. So, the main diagonal of matrix 𝐴 is the three entries one, zero, and three, and these are not all equal to one. So, matrix 𝐴 is not an identity matrix. However, it could be useful to finish off the rest of our identity matrix. All of the entries not on the main diagonal of our identity matrix need to be equal to zero. And of course, we can also see this is not true for the matrix 𝐴. The element in row three, column one is equal to one.
So, let’s instead move on to option (B). We need to check if matrix 𝐴 is a row matrix. And we recall we call a matrix a row matrix if it only has one row. However, we’ve already determined the number of rows of matrix 𝐴. Matrix 𝐴 has three rows. Therefore, matrix 𝐴 is not a row matrix.
So, let’s move on to option (C). We need to check if matrix 𝐴 is a diagonal matrix. And to help us define our diagonal matrix, let’s once again look at the identity matrix of order 𝑛. We define this matrix as a square matrix where every element whose row number is equal to its column number is equal to one. This is sometimes called the main diagonal of the matrix. And we define a diagonal matrix to be a matrix where every element which is not on the main diagonal is equal to zero. So, for example, the identity matrix of order 𝑛 is an 𝑛-by-𝑛 diagonal matrix, where every entry on the main diagonal is one.
So, to show that matrix 𝐴 is not diagonal, we just need to find an element not on its main diagonal, which is nonzero. And we already found this element when we showed that matrix 𝐴 was not an identity matrix. The element in row one, column three is equal to one. For 𝐴 to be a diagonal matrix, this would need to be zero.
Let’s now go through options (D) and (E). We need to determine if matrix 𝐴 is an upper or lower triangular matrix. For a matrix to be a lower triangular matrix, we recall all of the entries above the main diagonal must be equal to zero. Similarly, for a matrix to be an upper triangular matrix, all of the entries below the main diagonal must be equal to zero. And this means we can immediately eliminate option (E) because, remember, we’ve already shown the element in row one, column three is not equal to zero. This is below the main diagonal. So, matrix 𝐴 cannot be an upper triangular matrix. However, if we look at the three elements above the main diagonal, we can see these are all equal to zero.
Therefore, since every element above the main diagonal of matrix 𝐴 is equal to zero, we were able to show that matrix 𝐴 is a lower triangular matrix, which was option (D).