Scalar Multiplication of Matrices

There are many ways to multiply matrices. We can multiply them by other matrices, identity matrices, determinants, and much more. But what happens when we multiply a matrix by a real number? Is this even possible? Why would we want to do this? Let''s find out:

Scalar multiplication explained

When we multiply a matrix by a real number, we call this "scalar multiplication." The real number in this operation is called a "scalar." When we carry out this operation, we multiply each element in our matrix by the given scalar. Remember that elements are the numbers that lie inside the matrix.

We can write this as:

$A=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$

And for the matrix $r\times A$ :

$rA=\left[\begin{array}{cc}r{a}_{11}& r{a}_{12}\\ r{a}_{21}& r{a}_{22}\end{array}\right]$

As we can see, the concept is really quite simple. If we can do basic arithmetic multiplication, scalar multiplication shouldn''t be too difficult.

Practicing scalar multiplication

Now let''s try a scalar multiplication operation using the skills that we have just learned. Consider the following matrix:

$A=\left[\begin{array}{cc}2& 1\\ 3& -2\end{array}\right]$

What happens if we multiply this matrix by the scalar 4? In other words, what is the value of 4A?

$4A=4\left[\begin{array}{cc}2& 1\\ 3& -2\end{array}\right]$

We need to multiply each element in our matrix by the scalar 4:

$\left[\begin{array}{cc}4\times 2& 4\times 1\\ 4\times 3& 4\times \left(-2\right)\end{array}\right]$

We are left with:

$\left[\begin{array}{cc}8& 4\\ 12& -8\end{array}\right]$

Properties of scalar multiplication

As we can see, scalar multiplication operates based on the same basic principles of arithmetic multiplication. However, there are a few rules or "properties" that we must keep in mind. In this set of rules, we can assume that A and B are $m\times n$ matrices. $0_{m\times n}$ represents the $m\times n$ zero matrices, while p and q both represent scalars:

• Associative property: The associative property states that $p\times \left(qA\right)=\left(pq\right)\times A$ . This means you can first multiply the scalar $q$ with matrix $A$ , and then multiply the result by $p$ , or multiply $p$ and $q$ together first, and then multiply the result by matrix $A$ . Either way, you'll get the same answer.
• Closure property: According to the closure property, if you multiply a matrix $A$ by a scalar $p$ , the result will also be an $m\times n$ matrix.
• Commutative property: For scalar multiplication, the commutative property states that $pA=Ap$ . This means you can multiply the matrix $p$ by scalar $p$ , or the scalar $p$ by matrix $A$ . In either case, you'll get the same result. Note, however, that this property typically does not apply when multiplying two matrices together.
• Distributive property: The distributive property for scalar multiplication over matrix addition states that $\left(p+q\right)A=pA+qA$ and $p\left(A+B\right)=pA+pB$ . These rules are similar to the distributive property in standard arithmetic and algebra.
• Identity property: The identity property of scalar multiplication states that $1\times A=A$ . When a matrix is multiplied by one, the matrix remains unchanged.
• Multiplicative property of -1: The multiplicative property of -1 states that $\left(-1\right)\times A=-A$ . Multiplying a matrix by -1 results in a matrix where all the entries are negated.
• Multiplicative property of 0: The multiplicative property of 0 states that $0\times A=0_{m\times n}$ . This means that when a matrix is multiplied by zero, the result is a zero matrix of the same size. This rule mirrors the property of zero in arithmetic.

Matrix multiplication vs. scalar multiplication

Technically speaking, scalar multiplication and matrix multiplication are two different things. Although both operations involve multiplying matrices, there are different rules and properties that apply.

Scalar multiplication involves multiplying a matrix by a scalar (a single real or complex number). Each entry of the matrix is multiplied by this scalar. The operation is straightforward and shares many properties with ordinary number multiplication, such as associativity, commutativity, and distributivity.

Matrix multiplication, on the other hand, entails multiplying two matrices together. This operation is more complex and involves a series of multiplications and additions between the elements of the matrices. Furthermore, matrix multiplication doesn''t always obey the same rules as scalar multiplication or standard arithmetic multiplication. Notably, matrix multiplication is not generally commutative, meaning that the order in which you multiply matrices matters: for two matrices $A$ and $B$ , it''s not always the case that $AB=BA$ .

It''s also worth mentioning that many operations in linear algebra and matrix theory involve both scalar and matrix multiplications. For example, when calculating the inverse of a matrix, one may multiply a scalar such as $\frac{1}{\mathrm{determinant}}$ by a matrix.

Understanding both scalar multiplication and matrix multiplication is key to a range of applications, from solving systems of linear equations to transformations in computer graphics and data analysis in statistics.

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Scalar multiplication involves many familiar concepts, but it also requires students to learn new rules and properties. A smart way to memorize these new concepts is to work alongside a tutor in a 1-on-1 environment. A tutor can help your student memorize concepts using strategies that match their learning style. For example, a verbal learner can memorize the properties of scalar multiplication through rhymes and acronyms, while a visual learner can memorize them with flashcards. Tutors can also answer questions that your student might not have had the chance to ask during class. Reach out to our Educational Directors today to learn more, and remember: Varsity Tutors can match your student with a suitable math professional.