Explainer: Scalar Multiplication of Matrices

In this explainer, we will learn how to carry out scalar multiplication of matrices.

There are some operations in linear algebra, such as matrix transposition and matrix multiplication, which are defined in an intricate way that has little obvious resemblance to operations that are defined in conventional algebra. There are also special types of matrices, such as square matrices, which have more advanced concepts associated with them, such as the matrix determinant and the matrix inverse. Many of these concepts took a great deal of time to evolve since the idea of linear algebra was first considered (in an indirect sense) around the year 300 BC. For example, the first instance of the matrix determinant was recorded in the year 1683, approximately 2,000 years later!

The various concepts in linear algebra all have a perfectly sensible and justified motivation, even if some of them are slightly difficult to understand upon first reading. That being said, there are several concepts in linear algebra that are both well justified and simple to define. As well as the addition or subtraction of two matrices, another such example is the scalar multiplication of one matrix. Please note that the concept of scalar multiplication is different from the concept of matrix multiplication, as this latter idea is more intricate, with scalar multiplication being by far the easier of the two operations to understand.

Definition: Scalar Multiplication

For a matrix of order ๐‘šร—๐‘›, defined as ๐ด=โŽกโŽขโŽขโŽฃ๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘Ž๐‘Žโ‹ฏ๐‘Žโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Ž๐‘Žโ‹ฏ๐‘ŽโŽคโŽฅโŽฅโŽฆ,๏Šง๏Šง๏Šง๏Šจ๏Šง๏Š๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏Š๏‰๏Šง๏‰๏Šจ๏‰๏Š we can complete the โ€œscalar multiplicationโ€ by a number ๐‘˜. This requires multiplying every entry of ๐ด by ๐‘˜: ๐‘˜๐ด=โŽกโŽขโŽขโŽขโŽฃ๐‘˜ร—๐‘Ž๐‘˜ร—๐‘Žโ‹ฏ๐‘˜ร—๐‘Ž๐‘˜ร—๐‘Ž๐‘˜ร—๐‘Žโ€ฆ๐‘˜ร—๐‘Žโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘˜ร—๐‘Ž๐‘˜ร—๐‘Žโ‹ฏ๐‘˜ร—๐‘ŽโŽคโŽฅโŽฅโŽฅโŽฆ.๏Šง๏Šง๏Šง๏Šจ๏Šง๏Š๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏Š๏‰๏Šง๏‰๏Šจ๏‰๏Š This operation is often referred to as โ€œscalingโ€ the matrix by the constant ๐‘˜.

As an example, we consider the matrix ๐ด=โŽกโŽขโŽขโŽฃ53โˆ’1โˆ’311061754โˆ’27โˆ’26โˆ’26โˆ’38โŽคโŽฅโŽฅโŽฆ.

Suppose now that we wished to scale the matrix by the constant โˆ’2. We would then multiply every entry of ๐ด by โˆ’2, hence finding that โˆ’2๐ด=โŽกโŽขโŽขโŽขโŽฃ(โˆ’2)ร—5(โˆ’2)ร—3(โˆ’2)ร—(โˆ’1)(โˆ’2)ร—(โˆ’3)(โˆ’2)ร—1(โˆ’2)ร—1(โˆ’2)ร—0(โˆ’2)ร—6(โˆ’2)ร—1(โˆ’2)ร—7(โˆ’2)ร—5(โˆ’2)ร—4(โˆ’2)ร—(โˆ’2)(โˆ’2)ร—7(โˆ’2)ร—(โˆ’2)(โˆ’2)ร—6(โˆ’2)ร—(โˆ’2)(โˆ’2)ร—6(โˆ’2)ร—(โˆ’3)(โˆ’2)ร—8โŽคโŽฅโŽฅโŽฅโŽฆ.

Completing the operation for every entry gives โˆ’2๐ด=โŽกโŽขโŽขโŽฃโˆ’10โˆ’626โˆ’2โˆ’20โˆ’12โˆ’2โˆ’14โˆ’10โˆ’84โˆ’144โˆ’124โˆ’126โˆ’16โŽคโŽฅโŽฅโŽฆ.

It is not the case that we are restricted to scaling a matrix by an integer and we could equally choose to scale by a fraction, an irrational number, or even a complex number if we are feeling adventurous. Although it is not strictly necessary, if scaling a matrix by a fraction, it is normally considered good practice to simplify any resultant fractions into their simplest form. For example, we define the matrix ๐ต=๏˜5โˆ’36488306โˆ’11โˆ’1๏ค and decide to scale by the constant 13. We would find that 13๐ต=โŽกโŽขโŽขโŽขโŽขโŽขโŽฃ13ร—513ร—(โˆ’3)13ร—613ร—413ร—813ร—813ร—313ร—013ร—613ร—(โˆ’1)13ร—113ร—(โˆ’1)โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.

Reducing as many fractions as possible to their lowest form gives 13๐ต=โŽกโŽขโŽขโŽขโŽขโŽขโŽฃ53โˆ’12438383102โˆ’1313โˆ’13โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ.

Before discussing some challenging problems which arise from the definition of scalar multiplication, we will first practice one question.

Example 1: Multiplying a Matrix by a Scalar

Given that ๐ด=[โˆ’1โˆ’8], find 3๐ด.

Answer

To scale ๐ด by the given constant, we multiply every entry by this number and therefore we have 3๐ด=[3ร—(โˆ’1)3ร—(โˆ’8)]=[โˆ’3โˆ’24].

One of the key principles of scalar multiplication is that every single entry has the exact same process applied to it, namely, that every entry is multiplied by the same number. It is never the case that scalar multiplication multiplies different entries by different numbers. The following question gives an example of how this principle can be applied in terms of solving problems in linear algebra.

Example 2: Finding a Scalar Multiple of a Matrix

Given that ๐‘ฅร—๏”โˆ’20โˆ’3โˆ’5๏ =๏”1402135๏ , find the value of ๐‘ฅ.

Answer

By multiplying every entry by ๐‘ฅ in the left-hand matrix, we are looking to find ๐‘ฅ which solves the following equation: ๏–๐‘ฅร—(โˆ’2)๐‘ฅร—0๐‘ฅร—(โˆ’3)๐‘ฅร—(โˆ’5)๏ข=๏”1402135๏ , which can equivalently be written as ๏”โˆ’2๐‘ฅ0โˆ’3๐‘ฅโˆ’5๐‘ฅ๏ =๏”1402135๏ .

For two matrices to be equal, every single entry must be identical. We must therefore match the pairs of highlighted entries as shown below: ๏”โˆ’2๐‘ฅ0โˆ’3๐‘ฅโˆ’5๐‘ฅ๏ =๏”1402135๏ .

This produces the system of linear equations โˆ’2๐‘ฅ=14,0=0,โˆ’3๐‘ฅ=21,โˆ’5๐‘ฅ=35.

Apart from the trivial second equation, which is clearly true, we observe that every equation is solved by setting ๐‘ฅ=โˆ’7.

Often, when working with a matrix of interest, we would choose to factor out a scalar multiple from each entry, if possible. For example, if we were given the matrix ๐ด=๏˜36240โˆ’416โˆ’8โˆ’44โˆ’4๏ค, it is easily observed that this can be written as ๐ด=๏›4ร—(9)4ร—(6)4ร—(0)4ร—(โˆ’1)4ร—(4)4ร—(โˆ’2)4ร—(โˆ’1)4ร—(1)4ร—(โˆ’1)๏ง.

And therefore each entry has a factor of 4 that can be removed, thus obtaining ๐ด=4๏˜960โˆ’14โˆ’2โˆ’11โˆ’1๏ค.

In some situations, it may actually be preferable to define a new matrix ๐ต=๏˜960โˆ’14โˆ’2โˆ’11โˆ’1๏ค, thereby allowing us to write ๐ด=4๐ต.

Example 3: Scalar Multiplication

Given the matrix ๐ด=๏˜1โˆ’1652โˆ’4โˆ’3โˆ’174๏ค, what is the greatest number ๐‘˜ for which no entry of ๐‘˜๐ด is greater than 1?

Answer

Given the matrix ๐ด as defined above, we know that ๐‘˜๐ด=๏™๐‘˜โˆ’16๐‘˜5๐‘˜2๐‘˜โˆ’4๐‘˜โˆ’3๐‘˜โˆ’๐‘˜7๐‘˜4๐‘˜๏ฅ.

First, examining the entries where ๐‘˜ is multiplied by a positive number, we have ๐‘˜, 2๐‘˜, 4๐‘˜, 5๐‘˜, and 7๐‘˜. For none of these values to be greater than 1, we would require that ๐‘˜โ‰ค17. Since this value of ๐‘˜ is positive, the remaining values โˆ’๐‘˜,โˆ’3๐‘˜,โˆ’4๐‘˜, and โˆ’16๐‘˜ would all be negative and hence less than 1.

If we now focus on the entries โˆ’๐‘˜, โˆ’3๐‘˜, โˆ’4๐‘˜, and โˆ’16๐‘˜, we can see that these will all be less than 1 providing that ๐‘˜โ‰ฅโˆ’116. This value of ๐‘˜ is less than the restriction of ๐‘˜โ‰ค17 that we found above. Since we are searching for an upper limit on the value of ๐‘˜, the only possible answer is therefore ๐‘˜=17.

Scalar multiplication is an operation that features regularly in linear algebra. Along with addition, it is perhaps the most simple algebraic operation to understand. However, this does not mean that a problem in linear algebra can be thought of as simple, purely because it involves scalar multiplication. The following two questions will illustrate how scalar multiplication of matrices can provide rich and interesting examples that allow for a better level of comprehension to be developed.

Example 4: Solving Equations Involving Scalar Multiplication

Consider the matrix equation ๏”819โˆ’3๏ =๐‘š๏”302โˆ’1๏ +๏”โˆ’1130๏ .

Find the value of ๐‘š which solves this equation.

Answer

First, we rewrite the equation after incorporating the scalar multiplication by ๐‘š: ๏”819โˆ’3๏ =๏”3๐‘š02๐‘šโˆ’๐‘š๏ +๏”โˆ’1130๏ .

We complete the addition on the right-hand side of this equation, working entry by entry to find ๏”819โˆ’3๏ =๏”3๐‘šโˆ’112๐‘š+3โˆ’๐‘š๏ .

For the two matrices to be equal, it must be the case that the entries match exactly, as shown: ๏”819โˆ’3๏ =๏”3๐‘šโˆ’112๐‘š+3โˆ’๐‘š๏ .

This gives the system of linear equations 8=3๐‘šโˆ’1,1=1,9=2๐‘š+3,โˆ’3=โˆ’๐‘š.

The final equation gives ๐‘š=3 and it can be checked that all of the given equations are also true if ๐‘š=3, which means that this must be the answer. This can be checked by substituting ๐‘š=3 back into the original matrix equation and then observing that both sides of the equation match.

Example 5: Solving Equations Involving Scalar Multiplication

Find the numbers ๐‘Ž,๐‘, and ๐‘ so that ๐‘Ž๏”110โˆ’1๏ +๐‘๏”1001๏ +๐‘๏”0โˆ’110๏ =๏”10โˆ’13๏ .

Answer

We begin by incorporating the scalar terms into the matrices, giving ๏“๐‘Ž๐‘Ž0โˆ’๐‘Ž๏Ÿ+๏”๐‘00๐‘๏ +๏”0โˆ’๐‘๐‘0๏ =๏”10โˆ’13๏ .

Since matrix addition is completed entry by entry, we have ๏”๐‘Ž+๐‘๐‘Žโˆ’๐‘๐‘โˆ’๐‘Ž+๐‘๏ =๏”10โˆ’13๏ .

Given the highlighted version ๏”๐‘Ž+๐‘๐‘Žโˆ’๐‘๐‘โˆ’๐‘Ž+๐‘๏ =๏”10โˆ’13๏ , we must solve the system of linear equations ๐‘Ž+๐‘=1,๐‘Žโˆ’๐‘=0,๐‘=โˆ’1,โˆ’๐‘Ž+๐‘=3.

The third equation gives ๐‘=โˆ’1, which can be substituted into the second equation to show that ๐‘Ž=โˆ’1. Then, substituting the value of ๐‘Ž into either the first or the fourth equation gives that ๐‘=2.

Scalar multiplication has many attractive properties when combined with matrix addition. If we were working with conventional algebra then we know that the quantities ๐‘Ž,๐‘, and ๐‘ will always obey the rule ๐‘Žร—(๐‘+๐‘)=๐‘Žร—๐‘+๐‘Žร—๐‘, which is known as the โ€œdistributive property.โ€ It transpires that the same property holds for matrix addition and scalar multiplication.

Theorem: Distributive Property

Scalar multiplication is โ€œdistributiveโ€ when combined with matrix addition. In other words, assuming that ๐‘Ž is a scalar constant and that ๐ต and ๐ถ are matrices with the same order, then ๐‘Ž(๐ต+๐ถ)=๐‘Ž๐ต+๐‘Ž๐ถ.

We demonstrate this result by way of example. Let us set ๐‘Ž=3 and define the two matrices ๐ต=๏”โˆ’460872๏ ,๐ถ=๏”37โˆ’566โˆ’1๏ .

Then, ๐ต+๐ถ=๏”โˆ’460872๏ +๏”37โˆ’566โˆ’1๏ =๏”โˆ’113โˆ’514131๏ ; and hence ๐‘Ž(๐ต+๐ถ)=3๏”โˆ’113โˆ’514131๏ =๏”โˆ’339โˆ’1542393๏ .

Equally, we could have chosen a different route and first calculated ๐‘Ž๐ต=๏”โˆ’1218024216๏ ,๐‘Ž๐ถ=๏”921โˆ’151818โˆ’3๏ .

From this we could then have calculated that ๐‘Ž๐ต+๐‘Ž๐ถ=๏”โˆ’1218024216๏ +๏”921โˆ’151818โˆ’3๏ =๏”โˆ’339โˆ’1542393๏ .

Thus, we have shown in this example that ๐‘Ž(๐ต+๐ถ)=๐‘Ž๐ต+๐‘Ž๐ถ. Naturally, it is possible to prove the above theorem rigorously and without reference to any particular example, although here we have simply given a demonstration.

Although it may seem as though scalar multiplication is a trivial matrix operation, being able to work fluently with this concept is often the difference between the solution to a problem being short and simple and the solution being long and complicated. When examining any particular matrix, it is usually a good idea to check whether any scalar multiple can be easily removed from all entries. Many other concepts in linear algebra have at least some interaction with scalar multiplication, with key concepts like the determinant and the matrix multiplication being particularly notable.

Key Points

  1. Multiplying a matrix ๐ด by a scalar ๐‘˜ means that every entry of the matrix ๐ด will be multiplied by ๐‘˜.
  2. When multiplying a matrix ๐ด by a scalar ๐‘˜, we can also choose to write this as ๐‘˜๐ด, rather than completing the calculation described in the previous point.
  3. Scalar multiplication is distributive when combined with matrix addition.

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