Types of matrices

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Following are the types of a matrix.

Zero matrix

A zero matrix, denoted as $O$, is a matrix where all elements are zero. Adding a zero matrix to any matrix results in the same matrix.

Example

Singleton matrix

A singleton matrix is a matrix with a single element.

Example

Rectangular matrix

A rectangular matrix has different numbers of rows and columns.

Example

Square matrices

A square matrix has an equal number of rows and columns.

Example

Equal matrices

Two matrices are equal if they have the same dimensions and corresponding elements are equal.

Example

Diagonal matrix

A diagonal matrix has non-diagonal elements equal to zero.

Example

Triangular matrices (upper and lower)

A lower triangular matrix has all non-diagonal elements above the main diagonal equal to zero. In contrast, an upper triangular matrix has non-diagonal elements below the main diagonal equal to zero.

Example

Scalar matrix

A scalar matrix is a diagonal matrix where all diagonal elements are the same scalar value.

Example

Identity matrix

An identity matrix, denoted as $I$ is a square matrix with ones on the diagonal and zeros elsewhere.

Example

Symmetric and skew-symmetric matrices

A symmetric matrix is equal to its transpose, while a skew-symmetric matrix is one whose transpose is the negation of itself.

Example

Invertible matrices

An invertible matrix, also called a non-singular matrix, has a unique matrix inverse that, when multiplied, yields the identity matrix.

Example

Learn about orthogonal matrix.

Conclusion

Matrices play a fundamental role in mathematics and are widely applicable across various disciplines. Each matrix type holds unique properties that find practical use in different applications, from linear transformations and computer graphics to physics and engineering.