In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order ${\displaystyle n}$. Any two square matrices of the same order can be added and multiplied.

Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if ${\displaystyle R}$ is a square matrix representing a rotation (rotation matrix) and ${\displaystyle \mathbf {v} }$ is a column vector describing the position of a point in space, the product ${\displaystyle R\mathbf {v} }$ yields another column vector describing the position of that point after that rotation. If ${\displaystyle \mathbf {v} }$ is a row vector, the same transformation can be obtained using ${\displaystyle \mathbf {v} R^{\mathsf {T}}}$, where ${\displaystyle R^{\mathsf {T}}}$ is the transpose of ${\displaystyle R}$.

The entries ${\displaystyle a_{ii}}$ (i = 1, ..., n) form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements a₁₁ = 9, a₂₂ = 11, a₃₃ = 4, a₄₄ = 10.

The diagonal of a square matrix from the top right to the bottom left corner is called antidiagonal or counterdiagonal.

If all entries outside the main diagonal are zero, ${\displaystyle A}$ is called a diagonal matrix. If all entries below (resp. above) the main diagonal are zero, ${\displaystyle A}$ is called an upper (resp. lower) triangular matrix.

The identity matrix ${\displaystyle I_{n}}$ of size ${\displaystyle n}$ is the ${\displaystyle n\times n}$ matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g. $${\displaystyle I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ \ldots ,\ I_{n}={\begin{bmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{bmatrix}}.}$$ It is a square matrix of order ${\displaystyle n}$, and also a special kind of diagonal matrix. The term identity matrix refers to the property of matrix multiplication that $${\displaystyle I_{m}A=AI_{n}=A}$$ for any ${\displaystyle m\times n}$ matrix ${\displaystyle A}$.

A square matrix ${\displaystyle A}$ is called invertible or non-singular if there exists a matrix ${\displaystyle B}$ such that $${\displaystyle AB=BA=I_{n}.}$$ If ${\displaystyle B}$ exists, it is unique and is called the inverse matrix of ${\displaystyle A}$, denoted ${\displaystyle A^{-1}}$.

A square matrix ${\displaystyle A}$ that is equal to its transpose, i.e., ${\displaystyle A^{\mathsf {T}}=A}$, is a symmetric matrix. If instead ${\displaystyle A^{\mathsf {T}}=-A}$, then ${\displaystyle A}$ is called a skew-symmetric matrix.