Lesson Video: Types of Matrices Mathematics

Video Transcript

In this lesson, we’ll recap what we mean by a matrix before learning how to identify special types of matrices, such as square matrices, row, column, identity, zero, diagonal, lower triangular, and upper triangular matrices. A matrix is an array of numbers. And we arrange numbers or scalars, which we call elements, in a matrix in rows and columns. When we describe a matrix as 𝑚 by 𝑛, this is its order. And it tells us that it has 𝑚 rows and 𝑛 columns. Let’s take a matrix 𝐴. Its elements, defined as lowercase 𝑎 with this subscript, are as shown, where 𝑎 subscript 𝑖𝑗 is the element that appears in the 𝑖th row and 𝑗 column.

Now, one special type of matrix is one that occurs when 𝑚 is equal to 𝑛, in other words, where the number of rows is equal to the number of columns. And when this happens, we call it a square matrix. If not, it follows that the matrix is rectangular. Now, there are several other types of matrices, so let’s find out what they are.

Determine the type of matrix given by negative eight, two, negative seven, negative one. Is it (A) a unit matrix, (B) a column matrix, (C) a row matrix, or (D) a square matrix?

We know that if we take an 𝑚 by 𝑛 matrix, where 𝑚 is the number of rows and 𝑛 is the number of columns, a square matrix is such that 𝑚 is equal to 𝑛. In other words, the number of rows is equal to the number of columns. But what is the definition of our other three matrices? Well, firstly, the unit matrix is sometimes also known as the identity matrix. In an identity matrix, all of the elements are zero except the elements in the leading diagonal, which are one. Then a column matrix occurs when 𝑛 is equal to one. There is just one column. So it’s sometimes also called a vector.

Similarly, a row matrix occurs when 𝑚 is equal to one. There is just one row, and this is also called a vector. So if we look at our matrix negative eight, two, negative seven, negative one, which of these do we have? Firstly, we see that this cannot be an identity or a unit matrix. For this to be the case, the elements would need to be one, zero, zero, one. And they’re quite clearly aren’t. Similarly, neither 𝑛 nor 𝑚 is equal to one. There is more than one column and more than one row, so it’s neither a column matrix nor a row matrix. In fact, this is a two-by-two matrix. 𝑚 is equal to two, that’s the number of rows, as is 𝑛. These numbers are the same. We have the same number of rows and columns, and so the answer must be (D). It’s a square matrix.

Let’s consider another example.

Determine which of the following matrices is a column matrix. Is it (A) two, negative two, three, five? (B) Two, negative two, three. Is it (C) two, zero, zero, five? Is it (D) zero, zero, zero, zero? Or is it (E) two, negative two, three?

Let’s remind ourselves what we mean when we say that a matrix is a column matrix. We say that an 𝑚 by 𝑛 matrix or a matrix order 𝑚 by 𝑛 has 𝑚 rows and 𝑛 columns. Now, if 𝑛 is equal to one, we call this a column matrix. In other words, if the matrix only has one column, it’s a column matrix. So let’s find the order of each of our individual matrices. Matrix 𝐴 has two rows and two columns, so it’s a two-by-two matrix. Matrix 𝐵 has three rows and one column, so it’s a three by one. Matrix 𝐶 is again a two by two, as is matrix 𝐷. And then we have the matrix 𝐸, which has one row and three columns. So that’s one by three. Now, if we look carefully and we compare each of these to our definition, we see that the matrix which has a value of 𝑛 equal to one is the matrix 𝐵. And so the matrix which is a column matrix is indeed 𝐵.

Now, in fact, we’re also able to name the other types of matrices. When 𝑚 is equal to 𝑛, in other words, when the number of rows is equal to the number of columns, we say the matrix is square. So matrix 𝐴 is square, as is matrix 𝐶. Now, matrix 𝐷 is also square, but this is a special type of matrix in itself. Every single element in this matrix is zero. And when we have a square matrix where this is the case, we call this a null matrix or zero matrix. And then, finally, let’s consider the matrix 𝐸. This time, the value of 𝑚 is equal to one. There is simply one row, and so we call this a row matrix.

We’re now going to extend our definitions to include diagonal matrices, upper triangle, and lower triangle matrices.

If the matrix 𝐴 is equal to negative one, zero, zero, six, eight, zero, six, five, three, which of the following is true? (A) The matrix 𝐴 is an identity matrix. (B) The matrix 𝐴 is an upper triangle matrix. (C) The matrix 𝐴 is a lower triangle matrix. (D) The matrix 𝐴 is a diagonal matrix. Or (E) the matrix 𝐴 is a zero matrix.

Let’s go through each of our types of matrix and recall what they actually mean. An identity matrix is a square matrix where all the elements is zero except for the elements on the leading diagonal, which are equal to one. A three-by-three identity matrix, for instance, would look as shown with the elements one, zero, zero, zero, one, zero, zero, zero, one. Next, we have a zero matrix, sometimes known as a null matrix. This is a square matrix whose elements are all equal to zero. Now, in fact, if we compare the matrix 𝐴 with either of these definitions, we see that it cannot be an identity matrix, so we disregard (A). Nor can it be a zero matrix. And so, we’re going to disregard (E).

Let’s now consider options (B), (C), and (D). Now, a triangle matrix is a special type of square matrix. An upper triangle matrix is a square matrix where all the entries below the main diagonal are zero. Then we say it’s a lower triangle matrix if the entries above the main diagonal are zero. We recall that the main diagonal or the leading diagonal is this one. And so, do either of these two definitions hold? Well, yes, they do. The elements that sit above this diagonal are all equal to zero. And so this must be a lower triangle matrix. And so the answer is (C).

We will just double check the definition of (D). What does it mean for a matrix to be diagonal? Well, for a matrix to be diagonal, it must have entries below the main diagonal that are zero and above the main diagonal that are zero. Then the entries on the actual diagonal itself are not equal to zero. Of course, if we look carefully, we see this isn’t the case with matrix 𝐴. The elements that sit below the leading diagonal are not equal to zero. And so the answer is (C). The matrix 𝐴 is a lower triangle matrix.

Determine the type of the matrix given by 57, zero, zero, zero, negative 72, zero, zero, zero, zero. Is it (A) a row matrix, (B) an identity matrix, (C) a diagonal matrix, or (D) a column matrix?

Now, if we look carefully at these definitions, we see that we can disregard two of them immediately. We know that a row matrix is just as it sounds. It’s a matrix that consists of exactly one row. Our matrix, of course, has three rows, so it cannot be a row matrix. Similarly, a column matrix consists of exactly one column. And our matrix has three. So the answer cannot be (D). And so we have two left to choose from. We have the identity matrix and the diagonal matrix. Both of these matrices are special types of square matrices. We know that an identity matrix has all elements equal to zero except those in the main or leading diagonal.

And in this case, those have to be equal to one as shown. Then a diagonal matrix does look quite similar. All the elements above and below the main diagonal are equal to zero. And then we have a series of nonzero elements that only occur in the main diagonal. Now, not all of those elements needs to be nonzero. But we know that all of them can’t be zero because then we would have a null or zero matrix. And so, by comparing these definitions to our matrix, we see we can also disregard the identity matrix. We have 57 and negative 72. We can, however, say that all of the elements that sit above the main diagonal, that’s this, are zero and the elements that sit below it are zero. And so we have a diagonal matrix. And the answer is (C).

We’ll consider one more example.

If the matrix 𝐴 is equal to five, five, zero, which of the following is true? Is it (A) the matrix 𝐴 is a unit matrix? (B) The matrix 𝐴 is a diagonal matrix. (C) The matrix 𝐴 is a square matrix. (D) The matrix 𝐴 is a column matrix. Or (E) the matrix 𝐴 is a row matrix.

We recall that a matrix of order 𝑚 by 𝑛 has 𝑚 rows and 𝑛 columns. And if 𝑚 is equal to 𝑛 — in other words, the number of rows is equal to the number of columns — then the matrix is said to be square. So (C) is the square matrix. But we also know that both unit matrices and diagonal matrices must themselves be square. And so, let’s ask ourselves, is the matrix 𝐴 a square matrix? Well, it’s quite clear that it does not have the same number of rows and columns. In fact, it’s a one-by-three matrix. It has one row and three columns. And so since unit matrices and diagonal matrices are examples of square matrices, we’re able to disregard the first three options here. It cannot be any of these.

And so we’re left with two to choose from. We have column matrix and row matrix. Well, we know that a column matrix is such that 𝑛 is equal to one, we have one column, whereas a row matrix occurs when 𝑚 is equal to one, when there’s one row. Comparing the general form to the order of our matrix, and we can actually say that 𝑚 is equal to one and 𝑛 is equal to three. We have one row and three columns, so it cannot be a column matrix. But because 𝑚 is equal to one, it must be a row matrix.

We’ll now review the key points from this lesson. We know that an 𝑚 by 𝑛 matrix has 𝑚 rows and 𝑛 columns. And let’s say the element 𝑎 sub 𝑖𝑗 is the element in the 𝑖th row and 𝑗th column. The matrix is said to be square if 𝑚 is equal to 𝑛, if the number of rows is equal to the number of columns. If 𝑛 is equal to one, however, we have a column matrix. And if 𝑚 is equal to one, if there’s just one row, then the matrix is said to be a row matrix. We then say that the matrix is a unit or identity matrix if 𝑚 is equal to 𝑛, so it’s a square matrix, and all of the elements are equal to zero except those on the main or leading diagonal, which are one. In other words, for the element 𝑎 𝑖𝑗, if 𝑖 is equal to 𝑗, then that is one.

A matrix is said to be a null or zero matrix if all the elements are equal to zero. And we can also say that a matrix is diagonal if it’s square and all the elements are zero except those on the main diagonal. Now, of course, some elements on that main diagonal could be equal to zero, but not all of them, because then we’d have a null matrix. Similarly, if all of the elements on the main diagonal are equal to one, then we have an identity matrix. Finally, we define a triangular matrix as another special type of square matrix. It’s upper triangle if all entries below the main diagonal are zero and a lower triangle matrix if all elements above are zero.