Scalar multiplication is one of the simplest matrix operations. It involves multiplying a single number (called a scalar) by every element of a matrix.

Formula:

For a scalar \(c\) and a matrix \(A\), the resulting matrix \(B = cA\) is defined as:

\(B_{ij} = c \times A_{ij}\)

The resulting matrix will always have the same dimensions as the original matrix.

Example:

\( 3 \times \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3 \times 1 & 3 \times 2 \\ 3 \times 3 & 3 \times 4 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 9 & 12 \end{pmatrix} \)