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Scalar Multiplication of Matrices

Just as two or more real numbers or two or more matrices can be multiplied, did you know that it is possible to multiply a real number with a matrix or vice-versa? Multiplication of matrices generally falls into two categories, Scalar Matrix Multiplication, in which a single real number is multiplied with every other element of the matrix and Vector Matrix Multiplication wherein an entire matrix is multiplied by another one. In this topic, we will learn about the scalar multiplication of a matrix.

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What is Scalar Multiplication of Matrices?

Let’s understand the method with an example. Fatima has doubled the production at a factory A in all categories. Previously quantities (in standard units) produced by factory A were

Boys        Girls

$$\begin{bmatrix} 80 & 60 \\ 75 & 65\\ 90 & 85 \end{bmatrix}$$

Revised quantities produced by factory A are as given below:

Boys            Girls

$$\begin{bmatrix} 2 × 80 & 2 × 60 \\ 2 × 75 & 2 × 65\\ 2 × 90 & 2 × 85 \end{bmatrix}$$

This can be represented in the matrix form as

$$\begin{bmatrix} 160 & 120 \\ 150 & 130\\ 180 & 170 \end{bmatrix}$$

We observe that the new matrix is obtained by multiplying each element of the previous matrix by 2. In general, we may define multiplication of a matrix by a scalar as follows:

If A = [aij]m × n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k. In other words, kA = k [aij]m×n = [k (aij)]m×n, that is, (i, j)th element of kA is kaij for all possible values of i and j.

Scalars and Scalar Multiplication

When we work with matrices, we refer to real numbers as scalars. The term scalar multiplication refers to the product of a real number and a matrix. In scalar multiplication, each entry in the matrix is multiplied by the given scalar. For example, given that,

$$A = \begin{bmatrix} 10 & 6 \\ 4 & 3\end{bmatrix}$$

let’s find 2A

$\mathrm{To find2A, simply multiply each matrix entry by 2}$

$$2A = 2 . \begin{bmatrix} 10 & 6 \\ 4 & 3\end{bmatrix}$$

$$= \begin{bmatrix} 2 . 10 & 2 . 6 \\ 2 . 4 & 2 . 3\end{bmatrix}$$

$$= \begin{bmatrix} 20 & 12 \\ 8 & 6\end{bmatrix}$$

Properties of Scalar Multiplication of a Matrix

If A = [aij] and B = [bij] be two matrices of the same order, say m × n, and k and l are scalars, then

• k(A + B) = kA + kB,
(k + l)A = kA + lA
• k (A + B) = k ([aij] + [bij])
= k[aij + bij] = [k(aij + bij)] = [(kaij) + (kbij)]
= [kaij] + [kbij] = k[aij] + k[bij] = kA + kB
• (k + l)A = (k + l)[aij]
= [(k + l)aij] + [kaij] + [laij] = k[aij] + l[aij] = kA + lA

Solved Examples For You

Question 1: For a matrix A : α(βA) = ? ( Where α and β are real numbers)

1. β)A
2. β(aA)
3. α2A
4. β2A

Answer : Option A or Option B. α and β both are simply the scalars.

Question 2: $$If A =\begin{bmatrix} -3 & -5 \\ -6 & 0\end{bmatrix},$$ A + B = 2I, Find B.

1. $$\begin{bmatrix} 1 & 2 \\ -6 & -3\end{bmatrix}$$
2. $$\begin{bmatrix} 3 & 5 \\ 3 & 6\end{bmatrix}$$
3. $$\begin{bmatrix} 5 & 5 \\ 6 & 2\end{bmatrix}$$
4. $$\begin{bmatrix} 1 & 2 \\ 3 & 2\end{bmatrix}$$

Answer : A + B = 2I. Therefore, $$\begin{bmatrix} -3 & -5 \\-6 & 0\end{bmatrix} + B = 2 ×\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}$$ Or, $$B = \begin{bmatrix} 2 & 0 \\ 0 & 2\end{bmatrix} – \begin{bmatrix} -3 & -5 \\-6 & 0\end{bmatrix}$$ So, $$B = \begin{bmatrix} 2 & 0 \\ 0 & 2\end{bmatrix} + \begin{bmatrix} 3 & 5 \\6 & 0\end{bmatrix}$$ Or, $$B = \begin{bmatrix} 5 & 5 \\ 6 & 2\end{bmatrix}$$

Question 3: What is scalar in a matrix?

Answer: It is adifferent kind of diagonal matrix in which equal valued elements are along the diagonal. Moreover, it is a quantity that is completely described by its magnitude. Some examples of scalar are density, volume, energy, speed, mass, and time.

Question 4: What is the difference between scalar and vector?

Answer: Scalar is a quantity that we can fully describe by a magnitude only. And it describes just a single number. Some of its examples are speed, volume, mass, etc. On the other hand, vector refers to a quantity that has both direction and magnitude.

Question 5: Can a matrix by a scalar?

Answer: Generally, matrices of the same dimension form a vector space. Also, we can add them to each other and multiply them by scalars. In addition, multiplying a matrix by a scalar multiple all of the entries by that scalar, although multiplying a matrix by a 1 × 1 matrix only makes sense if it is a 1 × n row matrix.

Question 6: Is distance a scalar or a vector?

Answer: It refers to a quantity that tells how much area an object has covered during its motion therefore it is a scalar. However, displacement is a vector quantity.

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