Types of matrices

If you are wondering how may types of matrices are there, you are in the right place. Here you will find the most important types of matrices in linear algebra, and we explain each type of matrix with its definition and with examples so that you fully understand each one.

What are the different types of matrices?

In linear algebra the main types of matrices are:

  • Row matrix: matrix that only has one row.
  • Column matrix: matrix that only has one column.
  • Square matrix: matrix that has the same number of rows as columns.
  • Rectangular matrix: matrix whose number of rows is different from its number of columns.
  • Transpose of a matrix: special type of matrix that is obtained by changing the rows of the matrix for its columns.
  • Triangular matrix: matrix in which all elements above or below the main diagonal are zero.
  • Diagonal matrix: matrix in which all the entries outside the main diagonal are zero.
  • Scalar matrix: diagonal matrix in which all the values on the main diagonal are equal.
  • Identity matrix: scalar matrix whose values on the main diagonal are 1.
  • Null matrix: matrix which all its elements are equal to zero.
  • Symmetric matrix: matrix whose transpose is equal to the matrix itself.
  • Antisymmetric matrix: matrix whose transpose is equal to its negative.

Of course, there are more types of matrices, but these are the most used and the most important in mathematics. If you are interested in rare types of matrices, you can find any other special type of matrix in our website.

Once we’ve seen a brief definition of each type of matrix, now we are going to explain in detail all these types of matrices with their properties.

Row matrix

As its own name suggests, a row matrix has only one row. For example

  \displaystyle\begin{pmatrix} 3 & 6 & -2 \end{pmatrix}

Column matrix

A column matrix is a matrix that has a single column:

\displaystyle \begin{pmatrix} 6 \\[1.1ex] 4 \end{pmatrix}

Transpose of a matrix

The transpose of a matrix is the matrix obtained by changing its rows for its columns. This type of matrix is indicated by adding a “T” at the top right of the matrix \left(A ^T\right).

  \displaystyle A=  \begin{pmatrix} 2 & 3  \\[1.1ex] -1 & 5    \end{pmatrix}  \ \longrightarrow \ A^T= \begin{pmatrix} 2 & -1 \\[1.1ex] 3 & 5  \end{pmatrix}

 \displaystyle B= \begin{pmatrix} 1 & 5 & 4 \\[1.1ex] 3 & 0 & 2   \end{pmatrix}  \ \longrightarrow \ B^T= \begin{pmatrix} 1 & 3 \\[1.1ex] 5 & 0 \\[1.1ex] 4 & 2   \end{pmatrix}

Transposed matrices have several interesting properties, so we recommend you to see all the properties of the transpose of a matrix.

Square matrix

A square matrix is a matrix that has the same number of rows as columns.

For example, the following matrix is a square matrix of order 3:

\begin{pmatrix} 1 & 6 & 3 \\[1.1ex] 2 & 4 & 0 \\[1.1ex] 5 & -1 & 2 \end{pmatrix}

The main diagonal of a square matrix are the elements that go from the upper left corner to the lower right corner. On the other hand, the secondary diagonal of a square matrix are the elements that go from the lower left corner to the upper right corner.

We recommend you to see all the properties of square matrices, since they are the most used type of matrices and, therefore, they are very important for linear algebra.

Rectangular matrix

A rectangular matrix is the opposite of a square matrix, that is, its numbers of rows differs from its numbers of columns. For example:

\begin{pmatrix}3&5&-1&7 \\[1.1ex]0&4&1&2\end{pmatrix}

Triangular matrix

A triangular matrix is a matrix in which all the elements above or below the main diagonal are 0.

Triangular matrices are divided into two types: upper triangular matrices, whose elements below the main diagonal are zero, and lower triangular matrices, whose elements above the main diagonal are zero. To fully understand the differences between them, see more examples of triangular matrices.

Upper triangular matrix

 \displaystyle \begin{pmatrix} 4 & 1 & 7 \\[1.1ex] 0 & 2 & 5 \\[1.1ex] 0 & 0 & 3 \end{pmatrix}

Lower triangular matrix

  \displaystyle \begin{pmatrix} 1 & 0 & 0 \\[1.1ex] 2 & 3 & 0 \\[1.1ex] -1 & 2 & 4 \end{pmatrix}

Diagonal matrix

A diagonal matrix is a square matrix in which all elements that are not on the main diagonal are zeros. You can see all the properties and more examples of diagonal matrices in this link.

  \displaystyle \begin{pmatrix} 4 & 0 & 0 \\[1.1ex] 0 & 2 & 0 \\[1.1ex] 0 & 0 & -3 \end{pmatrix}

Although these matrices seem very simple because they have many zeros, they are actually very important for mathematics. In fact, there is a procedure to diagonalize a matrix, so diagonalizable matrices are of great importance.

Scalar matrix

A scalar matrix is a diagonal matrix in which all the elements of the main diagonal are equal.

  \displaystyle \begin{pmatrix} 4 & 0 & 0 \\[1.1ex] 0 & 4 & 0 \\[1.1ex] 0 & 0 & 4 \end{pmatrix}

Click here to see more examples of scalar matrices.

Identity matrix

The identity matrix, also called unit matrix, is a diagonal matrix in which all the elements of the main diagonal are 1.

  \displaystyle \begin{pmatrix} 1 & 0 & 0 \\[1.1ex] 0 & 1 & 0 \\[1.1ex] 0 & 0 & 1 \end{pmatrix}

The identity matrix is very useful to do operations since it is a matrix with only ones and zeros. Also, the identity matrix has a lot applications, for example it is used to calculate the inverse of a matrix. We recommend you to see all the properties of the identity matrix so that you can understand how useful it is.

Null matrix

A null matrix, also known as zero matrix, is a matrix in which all its elements are 0:

  \displaystyle \begin{pmatrix} 0 & 0 & 0 \\[1.1ex] 0 & 0 & 0 \\[1.1ex] 0 & 0 & 0 \end{pmatrix}

Although it may seem a useless matrix, it has its utility. Click here to see its applications and all null matrix properties.

Symmetric matrix

A symmetric matrix is a matrix in which the main diagonal is an axis of symmetry.

  \displaystyle \begin{pmatrix} 2 & 3 & -1 \\[1.1ex] 3 & 5 & 9 \\[1.1ex] -1 & 9 & 1 \end{pmatrix}

Due to the properties of symmetric matrices, the result of transposing a symmetric matrix is the matrix itself.

Antisymmetric matrix

An antisymmetric matrix, also known as skew-symmetric matrix, is a matrix in which the main diagonal is filled with zeros and is an axis of antisymmetry.

  \displaystyle \begin{pmatrix} 0 & 4 & 2 \\[1.1ex] -4 & 0 & -3 \\[1.1ex] -2 & 3 & 0 \end{pmatrix}

In the following link you can see all the properties of this type of matrix and more examples of antisymmetric matrices.

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