Scalar Multiplication with Matrices – Properties and Examples

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 

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