Concept

Square Matrix

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 .

A_{n \times n} = ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ a_{11} a_{21} ⋮ a_{n 1} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{n n} ⎦ ⎥ ⎥ ⎥ ⎥ ⎤

The entries

a_{11},

a_{22},

a_{33},

\dots,

a_{n n}

in a square matrix together form its main diagonal. The following matrices have an equal number of rows and columns, making them square matrices.

A_{2 \times 2} = [35 - 4 2] B_{3 \times 3} = ⎣ ⎢ ⎡ 06 - 12 - 1 - 2 9 3177 ⎦ ⎥ ⎤

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.

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