A scalar matrix is a special type of diagonal matrix where all the elements on the principal diagonal are equal, and all other elements are zero. This means that a scalar matrix is both a square matrix and a diagonal matrix¹².

Definition and Properties

A matrix ( A = [a_{ij}] ) of order ( n \times n ) is called a scalar matrix if:

• ( a_{ij} = 0 ) for ( i \neq j )

• ( a_{ij} = k ) for ( i = j ), where ( k ) is a constant

In simpler terms, all the diagonal elements are the same constant ( k ), and all off-diagonal elements are zero¹².

Examples

Here are some examples of scalar matrices:

• Order 1: [ \begin{bmatrix} 1 \end{bmatrix} ]

• Order 2: [ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} ]

• Order 3: [ \begin{bmatrix} 5 & 0 & 0 \ 0 & 5 & 0 \ 0 & 0 & 5 \end{bmatrix} ]

• Order 4: [ \begin{bmatrix} \sqrt{7} & 0 & 0 & 0 \ 0 & \sqrt{7} & 0 & 0 \ 0 & 0 & \sqrt{7} & 0 \ 0 & 0 & 0 & \sqrt{7} \end{bmatrix} ]

Key Characteristics