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Multiplying Matrices by Scalars

A scalar is a dimensionless number; matrices can be multiplied by scalars.

To multiply a matrix by a scalar, simply multiply each entry in the matrix by the value of the scalar.

Let's try it:

You might also see a problem that looks like this: $$\text{Find 3}A.$$

Right! We want $3$ times the matrix $A$, so our scalar is $3$. Let's find their product:

Scalar Multiplication: Example

Here, a scalar is a dimensionless number. This means we can multiply any matrix, regardless of its dimensions, by a scalar. To do so, you simply "distribute" the scalar:

As shown above, we distribute the scalar, $3$, to every entry in the matrix. Of course, these entries can be more complicated: And the scalar can be a variable as well:

(Video) Scalar Multiplication of Matrices

by Don't Memorise

This video by Don't Memorise explains what scalar multiplication is for matrices.

Summary

Scalar multiplication is when we multiply a matrix by a constant. For example,

$\begin{bmatrix} 2&4 \\ -5&1 \\ \end{bmatrix}$ $\cdot 3$

To multiply this scalar by the matrix, we multiply each element in the matrix with the constant.

$\begin{bmatrix} {2\cdot3}&{4\cdot3}\\ {-5\cdot3}&{1\cdot3} \\ \end{bmatrix}$

We can simplify to:

$\begin{bmatrix} {6}&{12}\\ {-15}&{3} \\ \end{bmatrix}$

The order of the resulting matrix, $2x2$, will be the same as the order of the original matrix, $2x2$.

We can also use this technique to find the negative of a given matrix. We can find the negative of a matrix by multiplying the matrix by a negative constant. We can use the technique we just learned.

$\begin{bmatrix} 1&3 \\ -5&2 \\ \end{bmatrix}$ $\cdot {-1}$

We can apply what we learned earlier,

$\begin{bmatrix} {1\cdot{-1}}&{3\cdot{-1}}\\ {-5\cdot{-1}}&{2\cdot{-1}} \\ \end{bmatrix}$

Simplifying this matrix, we get,

$\begin{bmatrix} -1&-3 \\ 5&-2 \\ \end{bmatrix}$

Multiply a Matrix by a Scalar

To multiply a matrix by a scalar, we multiply each entry in the matrix by the value of the scalar. This is called scalar multiplication.

The scalar needs to be a real number.

Basically, we are applying the distributive property to the values within the matrix. This concept is different from the multiplication of a matrix because we are multiplying the matrix by a scalar, not another matrix.

Example

$$2 \cdot \begin{bmatrix} 1&1&3 \\ 2&3&4 \\ \end{bmatrix} = \begin{bmatrix} 2&2&6 \\ 4&6&8 \\ \end{bmatrix}$$

Multiplying a matrix by a scalar always produces another matrix of the same size, illustrating the closure property.