Operations on Matrices>Scalar Multiplication

Scalar multiplication is a basic operation on matrices in which a single real number (scalar) multiplies every entry of a matrix.
This concept is used throughout higher mathematics, including systems of equations, transformations, and vector-space methods in physics and engineering.

Let \[A = [a_{ij}]_{m \times n}\] be a matrix and k be a real number (scalar).

Then the scalar multiple of A by k is the matrix kA defined as:

That is, each entry of A is multiplied by the scalar k.

The negative of a matrix A, denoted by -A, is defined as the scalar multiple \[-1 \cdot A\].

• So, if \[A = [a_{ij}]\], then \[-A = [-a_{ij}]\]

• Adding a matrix to its negative gives the zero matrix: A + (-A) = O

where O is the zero matrix of the same order as A.

Find the values of x and y from the following equation:

\[2 \begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix}+ \begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix}= \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}\]

Solution We have

\[2 \begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix}+ \begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix}= \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}\Rightarrow \begin{bmatrix} 2x & 10 \\ 14 & 2y-6 \end{bmatrix}+ \begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix}= \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}\]

or \[\begin{bmatrix} 2x+3 & 10-4 \\ 14+1 & 2y-6+2 \end{bmatrix}= \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}\Rightarrow \begin{bmatrix} 2x+3 & 6 \\ 15 & 2y-4 \end{bmatrix}= \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}\]

or 2x + 3 = 7 and 2y – 4 = 14 (Why?)

or 2x = 7 – 3 and 2y = 18

or \[x=\frac{4}{2}\] and \[y=\frac{18}{2}\]

 i.e. x = 2 and y = 9.

• Scalar multiplication: \[kA = [ka_{ij}]\].

• Negative of a matrix: -A = (-1)A.

• Order of matrix does not change after scalar multiplication.

• k(A + B) = kA + kB.

• (k + l)A = kA + lA.

• k(lA) = (kl)A.

• \[0 \cdot A = O\], \[1 \cdot A = A\].