Matrix Scalar Multiplication
This page follows the JoeCMath video Multiplying Matrices by a Number? It’s Easy! The video shows scalar multiplication by taking one number and applying it to every entry in the matrix.
Main Idea
To multiply a matrix by a scalar, multiply every entry in the matrix by that number.
If
\[A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix},\]then
\[kA= \begin{bmatrix} ka_{11} & ka_{12} \\ ka_{21} & ka_{22} \end{bmatrix}.\]The dimensions do not change. A $2\times 2$ matrix stays $2\times 2$, and a $3\times 3$ matrix stays $3\times 3$.
Watch this section: quick summary of scalar multiplication at 1:00.
Opening $2\times 2$ Example
The video starts with
\[A= \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}\]and asks what happens when the matrix is multiplied by $2$?
Each entry gets doubled:
\[\begin{aligned} 2A &= 2 \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \\ &= \begin{bmatrix} 2(1) & 2(3) \\ 2(2) & 2(4) \end{bmatrix} \\ &= \begin{bmatrix} 2 & 6 \\ 4 & 8 \end{bmatrix}. \end{aligned}\]The important part is that $2$ is applied to every entry.
Watch this section: the $2\times 2$ intro example at 0:00.
The $3\times 3$ Example
The main example in the video uses
\[A= \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 3 \\ 2 & 5 & 4 \end{bmatrix}\]and finds $5A$.
Start by multiplying the scalar $5$ into each entry:
\[5A= 5 \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 3 \\ 2 & 5 & 4 \end{bmatrix}.\]The first column becomes
\[5 \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix} = \begin{bmatrix} 5 \\ -5 \\ 10 \end{bmatrix}.\]The middle column becomes
\[5 \begin{bmatrix} 0 \\ 3 \\ 5 \end{bmatrix} = \begin{bmatrix} 0 \\ 15 \\ 25 \end{bmatrix}.\]The right column becomes
\[5 \begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 10 \\ 15 \\ 20 \end{bmatrix}.\]So the final matrix is
\[5A= \begin{bmatrix} 5 & 0 & 10 \\ -5 & 15 & 15 \\ 10 & 25 & 20 \end{bmatrix}.\]Watch this section: the $3\times 3$ matrix times $5$ example at 0:08.
What To Keep Straight
Scalar multiplication is different from multiplying two matrices together.
That means there is no row-by-column matching and no dimension compatibility check. You simply distribute the scalar through the matrix entries:
\[k \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} = \begin{bmatrix} ka & kb & kc \\ kd & ke & kf \end{bmatrix}.\]For a negative scalar, each entry changes sign as part of the multiplication. For a scalar of $0$, every entry becomes $0$.
Watch this section: scalar multiplication summary at 1:00.
Timestamp Guide
Related Linear Algebra Work
For nearby matrix operations, use the Matrix Addition and Subtraction companion and the Matrix Multiplication companion.
You can also browse the Matrices topic page or use the JoeCMath Matrix/Linear Algebra playlist.