A square matrix is a matrix with the same number of rows and columns. The dimension of a square matrix that has $n$ rows and $n$ columns is $n\times n.$ The entries $a_{11},$ $a_{22},$ $a_{33},$ $\dots ,$ $a_{nn}$ in a square matrix together form its main diagonal. The following matrices have an equal number of rows and columns, making them square matrices. Here, $A$ has $2$ rows and $2$ columns, making it a $2\times 2$ matrix with four elements. Some important properties of square matrices can be listed as follows.

• If the main diagonal elements in a square matrix are ones and the rest of the elements are zeros, then the matrix is an identity matrix.
• The sum of all main diagonal elements in a square matrix is called the trace of the matrix.
• The determinant is only defined for square matrices
• Only square matrices can be invertible.
• If a square matrix's determinant is zero, then it is not invertible and is called singular.
• Any two square matrices of the same dimension can be added, subtracted, and multiplied.