Definition of square matrix

Definition: A square matrix is a matrix with the same number of rows and columns. An $n \times n$ matrix is called a square matrix of order $n$.

$$A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}$$

Example:
$$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$
is a $2 \times 2$ square matrix.

Square matrices are particularly important because they:

• Have a well-defined determinant
• May have an inverse (if nonsingular)
• Represent linear operators from a vector space to itself
• Have eigenvalues and eigenvectors