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Types of Matrix,
Matrix Rules for Names



This page will identify the different matrix rules for matrix names that can be encountered in Math, along with showing some of the types of matrix that can be described by such matrix names.



Square Matrix

As the name suggests, a square matrix is simply a matrix that is square in shape. Where the number of rows and columns are the same.

The matrix   A  =  \begin{bmatrix} 4 & {\text{-}}7 & 2 \\ 3 & 0 & 1 \\ 0 & 2 & {\text{-}}4 \end{bmatrix}   from earlier is a square matrix. Featuring 3 rows and 3 columns.




Diagonal Matrix

A diagonal matrix is a square matrix where there are zeroes everywhere apart from the main diagonal from left to right.

The matrix   \begin{bmatrix} 4 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & {\text{-}}2 & 0 \\ 0 & 0 & 0 & 8 \end{bmatrix}   is an example of a diagonal matrix. Every element not on the main diagonal is zero.




Identity Matrix

The identity matrix in Math is a special kind of diagonal matrix.

Every element not on the main diagonal is still zero, but now each element on the diagonal is the number 1.

The notation for am identity matrix is  I.

I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}    and     I = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}   are both examples of an identity matrix.




Scalar Matrix

A scalar matrix is a diagonal matrix but each of the numbers on the diagonal are the same.

The matrix   \begin{bmatrix} 7 & 0 & 0 & 0 \\ 0 & 7 & 0 & 0 \\ 0 & 0 & 7 & 0 \\ 0 & 0 & 0 & 7 \end{bmatrix}   is a scalar matrix.




Triangular Matrix

A triangular matrix is a particular kind of square matrix, and there are two types.

A ‘lower triangular matrix’ is a matrix where every element above the main diagonal is zero.

Where as an ‘upper triangular matrix’ is a matrix where very element below the main diagonal is zero.


\begin{bmatrix} 7 & 0 & 0 & 0 \\ 3 & 8 & 0 & 0 \\ {\text{-}}1 & 9 & 3 & 0 \\ 5 & {\text{-}}4 & 2 & 7 \end{bmatrix}  is an upper triangular matrix.         \begin{bmatrix} 8 & 2 & 4 & {\text{-}}3 \\ 0 & 7 & 2 & 2 \\ 0 & 0 & 9 & {\text{-}}6 \\ 0 & 0 & 0 & 4 \end{bmatrix}  is a lower triangular matrix.




Symmetric Matrix

A symmetric matrix is a square matrix where the elements on both the upper and lower sides of the main diagonal are the same.

\begin{bmatrix} 5 & 7 & 3 \\ 7 & 2 & {\text{-}}8 \\ 3 & {\text{-}}8 & {\text{-}}1 \end{bmatrix}

It also turns out that a symmetric matrix is the same matrix when it’s transposed.  So  AT  =  A.




Zero Matrix

When learning about matrix names, a matrix that should also be mentioned is the zero matrix, sometimes referred to as the null matrix.

This is a matrix that consists only of zeroes, every element of the matrix is zero and there are no others.

\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}   and   \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}   are both zero matrices.

Zero matrices don’t have to be either square or any particular size.




Matrix Rules for Names, Hollow Matrix

A hollow matrix is sometimes referred to as a zero diagonal matrix.

It is a square matrix where every element in the main diagonal of the matrix is zero. The other elements of the matrix can be any number.

So it is the opposite of a diagonal matrix shown earlier.

\begin{bmatrix} 0 & 5 & {\text{-}}1 \\ 7 & 0 & 3 \\ 9 & 5 & 0 \end{bmatrix}   is a hollow matrix.






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