Scalar Multiplication of a Matrix

Scalar Multiplication of a Matrix

If A is a matrix of order m x n and k be a scalar (any complex number), then the scalar

multiple kA of A by k is defined to be a m x n matrix whose elements are obtained by

multiplying the corresponding elements of A by k .

In other words, if $A=[a_{ij}]_{m\times n}$

$then\ kA=[c_{ij}]_{m\times n} \ ,\ where c_{ij}=ka_{ij}$

For example : if

$A=\begin{bmatrix} 4 &7 &3 \\ 2&6 &8 \end{bmatrix} \ and \ k=7 \ , \ then$

$kA=7\begin{bmatrix} 4 &7 &3 \\ 2&6 &8 \end{bmatrix} = \begin{bmatrix} 7(4) &7(7) &7(3) \\ 7(2)&7(6) &7(8) \end{bmatrix} = \begin{bmatrix} 28 &49 &21 \\ 14&42 &56 \end{bmatrix} \ .$

If $A_{1},A_{2},A_{3}.....,A_{k}$ be all m x n matrices, then for numbers $\lambda _{1},\lambda _{2},\lambda _{3}.....,\lambda _{k}$ the
matrix is called a linear combination of the matrices $A_{1},A_{2},A_{3}.....,A_{k}$ .

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Scalar Multiplication of a Matrix

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