Lesson Explainer: Types of Matrices Mathematics

In this explainer, we will learn how to identify special types of matrices like square, row, column, identity, zero, diagonal, lower triangular, and upper triangular matrices.

When describing matrices in a general sense, we often prefer to use shortened notation rather than going through the potentially tedious task of manually writing out every single entry. This is especially true when we are working with matrices that have a large number of rows and columns. Given that a matrix of order ๐‘šร—๐‘› has ๐‘š rows and ๐‘› columns, this will have a total number of ๐‘š๐‘› entries. It is easy to see why writing out such a matrix in full may not be a very fulfilling task.

Provided that we do not need to specify the value of the entries in a matrix, it is often convenient to refer to matrices only by their order, thereby describing the number of rows and columns but without describing the entries that it contains. When considering general operations such as matrix addition or multiplication, knowing the order is essential to check whether or not the operation is valid to perform, before the individual entries are even considered.

Even though knowing the order of a matrix can give important, high-level information, there are other ways of categorizing matrices that can also help to guide calculations and determine which calculations will be valid. We therefore begin to subdivide matrices into different โ€œtypes,โ€ which are often given names containing geometric descriptions in accordance with their properties. It is entirely possible for one particular matrix to have more than one type, as we will see later. We begin this explainer by defining several types of matrices that occur repeatedly in linear algebra.

Definition: Row Matrices and Column Matrices

A matrix of order 1ร—๐‘› is referred to as a โ€œrow matrixโ€ and has the general form (๐‘Ž๐‘Žโ‹ฏ๐‘Ž),๏Šง๏Šง๏Šง๏Šจ๏Šง๏Š whereas a โ€œcolumn matrixโ€ is a matrix of order ๐‘šร—1 and has the general form โŽ›โŽœโŽœโŽ๐‘Ž๐‘Žโ‹ฎ๐‘ŽโŽžโŽŸโŽŸโŽ .๏Šง๏Šง๏Šจ๏Šง๏‰๏Šง

For example, the following matrices are all row matrices because they have only one row: ๐ด=(3โˆ’3),๐ต=(130),๐ถ=(0โˆ’1542), whereas the following matrices are not row matrices because they all have more than one row: ๐ท=๏€ผโˆ’312โˆ’32โˆ’2๏ˆ,๐ธ=โŽ›โŽœโŽœโŽ0344โŽžโŽŸโŽŸโŽ ,๐น=๏€45โˆ’308876โˆ’1๏Œ.

In the example above, the only column matrix is ๐ธ, whereas all of the other matrices have more than one column and so these are not column matrices.

Definition: Square Matrices

A โ€œsquareโ€ matrix is one which has ๐‘› rows and ๐‘› columns, having the general form โŽ›โŽœโŽœโŽ๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘Ž๐‘Žโ‹ฏ๐‘Žโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Ž๐‘Žโ‹ฏ๐‘ŽโŽžโŽŸโŽŸโŽ .๏Šง๏Šง๏Šง๏Šจ๏Šง๏Š๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏Š๏Š๏Šง๏Š๏Šจ๏Š๏Š

The order of such a matrix is ๐‘›ร—๐‘› and there will be ๐‘›๏Šจ entries in total.

Square matrices are highly important mathematical objects that are of regular consideration in linear algebra. Many deep, useful theorems are based on the involved matrices being square. Although the operations of matrix addition and matrix multiplication can be defined on matrices of a general order, the operations of exponentiation and inversion are only defined when the matrix is square.

As an example, the following matrices each have an equal number of rows and columns and hence are classified as square matrices: ๐ด=๏€ผโˆ’2โˆ’1โˆ’22๏ˆ,๐ต=โŽ›โŽœโŽœโŽ014โˆ’350โˆ’35โˆ’3โˆ’2253โˆ’30โˆ’2โŽžโŽŸโŽŸโŽ ,๐ถ=๏€45โˆ’308876โˆ’1๏Œ, whereas none of the following matrices are square: ๐ท=๏€ผ3โˆ’10012๏ˆ,๐ธ=โŽ›โŽœโŽœโŽ44โˆ’6โˆ’6โˆ’443โˆ’5โŽžโŽŸโŽŸโŽ ,๐น=(463).

Square matrices are so important that many other special types of matrices have a definition which first specifies that they are already a square matrix.

Definition: Diagonal Matrices

For a matrix ๐ด, the โ€œdiagonalโ€ entries are those which have the same row number and column number, and the โ€œnondiagonalโ€ entries are all other entries. If ๐ด is a square matrix and all of the nondiagonal entries are zero, then ๐ด is called a โ€œdiagonalโ€ matrix.

We have highlighted in red all of the diagonal entries below: ๐ด=๏€ผ300โˆ’3๏ˆ,๐ต=๏€โˆ’30000000โˆ’2๏Œ,๐ถ=โŽ›โŽœโŽœโŽ8000060000200007โŽžโŽŸโŽŸโŽ .

Given that the matrices are all square matrices and that all nondiagonal entries are zero, the above are all diagonal matrices. We should comment that the matrix ๐ต has the diagonal entry ๐‘=0๏Šจ๏Šจ. Even though the value of this diagonal entry is zero, this is consistent with the definition above, which only requires that the nondiagonal elements have a value of zero.

The following matrices are not diagonal: ๐ท=๏€ผ1โˆ’403๏ˆ,๐ธ=๏€3000430โˆ’21๏Œ,๐น=โŽ›โŽœโŽœโŽ400060006000โŽžโŽŸโŽŸโŽ .

We have highlighted in orange the nondiagonal entries in ๐ท and ๐ธ which are nonzero and hence prevent these matrices from being in diagonal form. The matrix ๐น could never have been a diagonal matrix, as it is not a square matrix and therefore does not meet the requirements of the definition above. Although the concept of a diagonal-rectangular matrix does exist, it is used far less frequently than the standard, square diagonal matrices and is therefore outside the scope of this explainer.

Definition: Upper and Lower Triangular Matrices

A square matrix whose entries on one side of the diagonal are zero is a โ€œtriangular matrix.โ€

If the entries below the diagonal are zero, it is an โ€œupper triangular matrix.โ€

If the entries above the diagonal are zero, it is a โ€œlower triangular matrix.โ€

In particular, we note that a diagonal matrix is both upper triangular and lower triangular since all off-diagonal entries are zeros. Matrix ๐ท above is upper triangular since the only entry below the diagonal is zero. Matrix ๐ธ above is not triangular because of the nonzero entries: โˆ’2 below the diagonal and 3 above the diagonal. Matrix ๐น is not a triangular matrix since it is not a square matrix.

Additionally, we consider the following three matrices: ๐‘‹=๏€10โˆ’1030001๏Œ,๐‘Œ=๏€000130631๏Œ,๐‘=๏€10โˆ’1030โˆ’101๏Œ.

We can see that ๐‘‹ is an upper triangular matrix since it is a square matrix where all entries below the main diagonal are equal to zero. ๐‘Œ is a lower triangular matrix since it is a square matrix where all entries above the main diagonal are equal to zero. On the other hand, ๐‘ is not a triangular matrix, since there are nonzero entries on both sides of the main diagonal.

Definition: Identity Matrices

An โ€œidentity matrix,โ€ sometimes called a โ€œunit matrix,โ€ is a diagonal matrix with ones on the main diagonal and zeros elsewhere. We often refer to these matrices by the special symbols 1๏Š or ๐ผ๏Š, where ๐‘› denotes that the matrix is of order ๐‘›ร—๐‘›.

Since an identity matrix is a diagonal matrix, it should be a square matrix.

The matrices below are examples of identity matrices: ๐ผ=๏€ผ1001๏ˆ,๐ผ=๏€100010001๏Œ,๐ผ=โŽ›โŽœโŽœโŽ1000010000100001โŽžโŽŸโŽŸโŽ ,๏Šจ๏Šฉ๏Šช whereas none of the previous matrices in this explainer have been identity matrices.

Definition: Zero Matrices

A โ€œzero matrixโ€ of order ๐‘šร—๐‘› is a matrix with ๐‘š rows and ๐‘› columns for which every entry is zero, which is often denoted as ๐‘‚๏‰๏Š. If a zero matrix is also a square matrix of order ๐‘›ร—๐‘›, then the notation ๐‘‚๏Š is sometimes used.

For example, the following matrices are all zero matrices: ๐‘‚=๏€ผ00000000๏ˆ,๐‘‚=๏€ผ0000๏ˆ,๐‘‚=๏€000000000๏Œ.๏Šจ๏Šช๏Šจ๏Šฉ

There are various other types of matrices that merit their own classification and label, such as symmetric matrices, skew-symmetric matrices, orthogonal matrices, block matrices, and so on. Each of these is interesting and rich enough to require its own, separate investigation.

Once the concepts above have been fully understood, it is possible to begin understanding these important types of matrices and their many applications. To practice recognizing these common types of matrices, in the following pages we give several questions that rely on the definitions that we have given above.

Example 1: Types of Matrices

If the matrix ๐ด=(550), which the following is true?

  1. The matrix ๐ด is a row matrix.
  2. The matrix ๐ด is a square matrix.
  3. The matrix ๐ด is a unit matrix.
  4. The matrix ๐ด is a column matrix.
  5. The matrix ๐ด is a diagonal matrix.

Answer

Let us begin by recalling the definition of each type of matrix listed in the options:

  1. A row matrix is a matrix with exactly one row.
  2. A square matrix is a matrix where the number of rows is equal to the number of columns.
  3. A unit matrix is a square matrix with 1s in the diagonal entries and 0s elsewhere.
  4. A column matrix is a matrix with exactly one column.
  5. A diagonal matrix is a square matrix where all entries outside the main diagonal are 0s.

This matrix is evidently not a square matrix since it does not have the same number of rows as columns, which eliminates option (B). This implies that the answer also cannot be option (C) because a unit matrix is a square matrix (in particular, a diagonal matrix with all diagonal entries having the same value of 1). This also implies that the answer cannot be option (E) since a diagonal matrix is a square matrix. Since the matrix has only one row and more than one column, it is a row matrix and hence the correct choice is (A).

Example 2: Types of Matrices

Determine the type of the matrix โŽ›โŽœโŽœโŽ4128โŽžโŽŸโŽŸโŽ .

  1. Row matrix
  2. Square matrix
  3. Identity matrix
  4. Column matrix

Answer

Let us begin by recalling the definition of each type of matrix listed in the options:

  1. A row matrix is a matrix with exactly one row.
  2. A square matrix is a matrix where the number of rows is equal to the number of columns.
  3. An identity matrix is a square matrix with 1s in the diagonal entries and 0s elsewhere.
  4. A column matrix is a matrix with exactly one column.

This matrix has more than one row and exactly one column, so it is a column matrix. The correct answer is option (D).

Example 3: Types of Matrices

Determine the type of the matrix ๏€ผโˆ’82โˆ’7โˆ’1๏ˆ.

  1. Row matrix
  2. Square matrix
  3. Unit matrix
  4. Column matrix

Answer

Let us begin by recalling the definition of each type of matrix listed in the options:

  1. A row matrix is a matrix with exactly one row.
  2. A square matrix is a matrix where the number of rows is equal to the number of columns.
  3. A unit matrix is a square matrix with 1s in the diagonal entries and 0s elsewhere.
  4. A column matrix is a matrix with exactly one column.

This matrix has more than one row and column and so cannot be either option (A) or option (D). Since the number of rows is the same as the number of columns, the matrix is a square matrix and so the answer is either option (B) or option (C). For a square matrix to be a unit matrix, it must also be a diagonal matrix, which is not true in this instance. Therefore, the answer cannot be option (C), which means that (B) is the correct answer.

Example 4: Types of Matrices

Determine the type of the matrix ๏€000000000๏Œ.

  1. Row matrix
  2. Zero matrix
  3. Identity matrix
  4. Column matrix

Answer

Let us begin by recalling the definition of each type of matrix listed in the options:

  1. A row matrix is a matrix with exactly one row.
  2. A zero matrix is a matrix where all entries are 0s.
  3. An identity matrix is a square matrix with 1s in the diagonal entries and 0s elsewhere.
  4. A column matrix is a matrix with exactly one column.

The matrix has more than one row and more than one column, so it is neither a row matrix nor column matrix, which excludes options (A) and (D). Although the matrix is a square matrix and is also actually a diagonal matrix (since all of the nondiagonal entries are zero), it cannot be an identity matrix because the diagonal entries are not all equal to 1. Therefore, the answer cannot be option (C). Since every entry of the matrix is zero, the correct answer is option (B). We would refer to the given matrix as the zero matrix ๐‘‚๏Šฉ๏Šฉ or, if we prefer the shorthand notation for square-zero matrices, we would simply refer to this as ๐‘‚๏Šฉ.

Example 5: Types of Matrices

If the matrix ๐ด=๏€ผ1001๏ˆ, which of the following is true?

  1. The matrix ๐ด is a row matrix.
  2. The matrix ๐ด is a zero matrix.
  3. The matrix ๐ด is an identity matrix.
  4. The matrix ๐ด is a column matrix.

Answer

Let us begin by recalling the definition of each type of matrix listed in the options:

  1. A row matrix is a matrix with exactly one row.
  2. A zero matrix is a matrix where all entries are 0s.
  3. An identity matrix is a square matrix with 1s in the diagonal entries and 0s elsewhere.
  4. A column matrix is a matrix with exactly one column.

The given matrix cannot be either of options (A) and (D). Given that there are nonzero entries in this matrix, we know that option (B) is not possible. The given matrix is a diagonal matrix and all of the diagonal entries are equal to 1, so this an identity matrix and would be denoted either as ๐ผ๏Šฉ or as 1๏Šฉ. Therefore, the correct answer is option (C).

Example 6: Types of Matrices

Determine the type of the matrix ๏€57000โˆ’720000๏Œ.

  1. Row matrix
  2. Diagonal matrix
  3. Identity matrix
  4. Column matrix

Answer

Let us begin by recalling the definition of each type of matrix listed in the options:

  1. A row matrix is a matrix with exactly one row.
  2. A diagonal matrix is a matrix where all entries outside the main diagonal are 0s.
  3. An identity matrix is a square matrix with 1s in the diagonal entries and 0s elsewhere.
  4. A column matrix is a matrix with exactly one column.

Clearly this matrix is neither a row matrix nor a column matrix, so options (A) and (D) cannot be correct. We can see that the dimension of this matrix is 3ร—3, which means that this matrix has the same number of rows as columns. This means that it is a square matrix. Since, additionally, all entries off the main diagonal are equal to zero, this is a diagonal matrix. Given that the diagonal entries are not all equal to 1, this cannot be an identity matrix and therefore option (C) is not possible. Note that even though one of the diagonal entries is equal to zero, this does not prevent the matrix from being classified as diagonal and thus the correct answer is (B).

Example 7: Identifying the Type of a Triangular Matrix

If the matrix ๐ด=๏€โˆ’100680653๏Œ, which of the following is true?

  1. The matrix ๐ด is an upper triangular matrix.
  2. The matrix ๐ด is a lower triangular matrix.
  3. The matrix ๐ด is an identity matrix.
  4. The matrix ๐ด is a diagonal matrix.
  5. The matrix ๐ด is a zero matrix.

Answer

Let us begin by recalling the definition of each type of matrix listed in the options:

  1. An upper triangular matrix is a matrix whose entries below the diagonal are 0s.
  2. A lower triangular matrix is a matrix whose entries above the diagonal are 0s.
  3. An identity matrix is a square matrix with 1s in the diagonal entries and 0s elsewhere.
  4. A diagonal matrix is a matrix where all entries outside the main diagonal are 0s.
  5. A zero matrix is a matrix where all entries are 0s.

In the given matrix, we can see that all entries above the diagonal are 0s. This means that the given matrix is a lower triangular matrix.

This is option (B).

With further study of linear algebra, it will become apparent why we have chosen to identify these particular types of matrices. Often, it is possible to gain significant high-level understanding of a problem just by understanding the types of matrices that are involved. For example, if we are working with a square matrix then we will actually be working with a matrix that is guaranteed to preserve dimensions when combined in a certain way with row matrices and column matrices. As another example, we mentioned above that a square matrix is necessary for the possibility of matrix โ€œdivisionโ€ (known properly as โ€œinversionโ€) to be well-defined.

In addition to the types of matrices defined above, there are other types of matrices that are of fundamental importance in physics and the other sciences. The concept of a โ€œHermitianโ€ matrix, for instance, is necessary for quantum mechanics to be well defined in a mathematical sense and therefore produce results that could be measured in real life.

Let us finish by recapping a few important definitions from this explainer.

Key Points

  • Row matrices and column matrices, respectively, have either only one row or one column.
  • Square matrices have an equal number of rows and columns.
  • Diagonal matrices are square matrices where every nondiagonal element has a value of zero.
  • A square matrix whose entries on one side of the diagonal are zero is a triangular matrix. If the entries below the diagonal are zero, it is an upper triangular matrix. If the entries above the diagonal are zero, it is a lower triangular matrix.
  • An identity matrix (often referred to as the unit matrix) is a diagonal matrix where the diagonal entries all have a value of 1. The notation for an identity matrix of order ๐‘›ร—๐‘› is either 1๏Š or ๐ผ๏Š and these matrices are of vital importance in linear algebra.
  • A zero matrix of order ๐‘šร—๐‘› is populated only by entries which have a value of zero. We often label such a matrix as ๐‘‚๏‰๏Š.

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