Here you will learn scalar multiplication with matrices (multiplicaition of a matrix by a scalar ) and properties of scalar mutiplication.
Let’s begin –
Scalar Multiplication with Matrices
Definition : Let \([a_{ij}]\) be an \(m\times n\) matrix and k be any number called a scalar. Then the matrix obtained by mutiplying every element of A by k is called the scalar multiple of A by k and is denoted by kA.
Thus,
kA = \([ka_{ij}]_{m\times n}\)
Example : if A = \(\begin{bmatrix} 1 & 2 & 5 \\ -2 & 3 & 4 \\ 1 & 2 & -1 \end{bmatrix}\), then 3A = \(\begin{bmatrix} 3 & 6 & 15 \\ -6 & 9 & 12 \\ 3 & 6 & -3 \end{bmatrix}\)
if A = \(\begin{bmatrix} 6 & 2 & 3 \\ 2 & 3 & -2 \\ 2 & 4 & 1 \end{bmatrix}\), then \(1\over 2\)A = \(\begin{bmatrix} 3 & 1 & 3/2 \\ 1 & 3/2 & -1 \\ 1 & 2 & 1/2 \end{bmatrix}\)
Example : Let A = \(\begin{bmatrix} 1 & 5 & 7 & 3\\ -1 & 5 & 9 & 4 \\ -2 & 6 & 3 & -5 \end{bmatrix}\), then 2A = \(\begin{bmatrix} 2 & 10 & 14 & 6\\ -2 & 10 & 18 & 8 \\ -4 & 12 & 6 & -10 \end{bmatrix}\)
and \(1\over 2\)A = \(\begin{bmatrix} 1/2 & 5/2 & 7/2 & 3/2 \\ -1/2 & 5/2 & 9/2 & 2 \\ -1 & 3 & 3/2 & -5/2 \end{bmatrix}\)
Properties of scalar multiplication
Various properties of scalar multiplication are stated below :
If A = \([a_{ij}]_{m\times n}\), B = \([b_{ij}]_{m\times n}\) are two matrices and k and l are scalars, then
(i) k (A + B) = k A + k B
(ii) (k + l) A = k A + l A
(iii) (k l) A = k (l A) = l (k A)
(iv) (-k) A = – (k A) = k(-A)
(v) 1 A = A
(vi) (-1) A = -A