Scalar Multiplication of Matrices

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

Revised quantities produced by factory A are as given below:

Boys            Girls

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

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

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

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Properties of Scalar Multiplication of a Matrix

If A = [a_ij] and B = [b_ij] 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 ([a_ij] + [b_ij])
  = k[a_ij + b_ij] = [k(a_ij + b_ij)] = [(ka_ij) + (kb_ij)]
  = [ka_ij] + [kb_ij] = k[a_ij] + k[b_ij] = kA + kB
• (k + l)A = (k + l)[a_ij]
  = [(k + l)a_ij] + [ka_ij] + [la_ij] = k[a_ij] + l[a_ij] = kA + lA

Solved Examples For You

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

1. (αβ)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.