Multiplication of a Matrix by a Scalar

Scalar Multiplication: The Core Idea

When a factory doubles its output, every entry in the production matrix is multiplied by 2. This operation — multiplying every element of a matrix by a fixed number — is called scalar multiplication. The fixed number is called a scalar (any real number).

Definition of Scalar Multiplication

Let $A = [a_{ij}]_{m \times n}$ be a matrix and let $k$ be any scalar. Then the product $kA$ is another matrix of the same order $m \times n$, defined by:

In words: multiply every element of $A$ by $k$. The $(i, j)$-th element of $kA$ is $k \cdot a_{ij}$ for every $i$ and $j$.

Worked Example

If $A = \begin{bmatrix} 3 & 1 & 1.5 \\ 5 & 7 & -3 \\ 2 & 0 & 5 \end{bmatrix}$, then $3A$ is:

Negative of a Matrix

The negative of a matrix $A$, denoted $-A$, is the scalar multiplication of $A$ by $-1$:

This simply means: change the sign of every element of $A$. If $A = \begin{bmatrix} 3 & 1 \\ -5 & x \end{bmatrix}$, then

Difference of Two Matrices

The difference $A - B$ of two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ of the same order $m \times n$ is defined as:

$A - B = A + (-1)B$ …