Scalar Multiplication

Let (mathbf{A}) by a (ntimes k) matrix and let (c) be a real number. That is, a real number is a (1times 1) matrix and is also called a scalar. Multiplying a scalar (c) by a matrix (mathbf{A}) is denoted by (cmathbf{A}) and defined by begin{equation*} cmathbf{A}=left( begin{array}{cccc} ca_{11} & ca_{12} & cdots & ca_{1k} \ vdots & vdots & ddots & vdots \ ca_{n1} & ca_{n2} & cdots & ca_{nk} end{array} right) . end{equation*} For example, suppose that (c=10) and begin{equation*} mathbf{A}=left( begin{array}{cc} 1 & 2 \ 6 & 8 end{array} right) text{ then }mathbf{B}=cmathbf{A}=left( begin{array}{cc} 10 & 20 \ 60 & 80 end{array} right) . end{equation*} Note that (cmathbf{A}=mathbf{A}c).