Matrix Scalar Multiplication

Matrix scalar multiplication is multiplying a matrix by a scalar. A scalar is a real number whereas a matrix is a rectangular array of numbers. When we deal with matrices, we come across two types of multiplications:

• Multiplying a matrix by another matrix and is called "matrix multiplication"
• Multiplying a matrix by a scalar (a number) and is called "matrix scalar multiplication"

Let us learn how to do matrix scalar multiplication and its properties along with examples.

The matrix scalar multiplication is the process of multiplying a matrix by a scalar. Let 'A' be a matrix and 'k' be a scalar (real number). Then kA is the result of the matrix scalar multiplication. To find kA, we just multiply every element of A by 'k'. Here are some examples.

Example: If A = \(\left[\begin{array}{ll}
-1 & 2 \\
0 & 3
\end{array}\right]\) then

• 2A = 2 \(\left[\begin{array}{ll}
  -1 & 2 \\ \\
  0 & 3
  \end{array}\right]\) = \(\left[\begin{array}{ll}
  2(-1) & 2(2) \\ \\
  2(0) & 2(3)
  \end{array}\right]\) = \(\left[\begin{array}{ll}
  -2 & 4 \\ \\
  0 & 6
  \end{array}\right]\)
• (1/2) A = (1/2) \(\left[\begin{array}{ll}
  -1 & 2 \\ \\
  0 & 3
  \end{array}\right]\) = (1/2)\(\left[\begin{array}{ll}
  (1/2)-1 & (1/2)2 \\ \\
  (1/2)0 & (1/2)3
  \end{array}\right]\) = \(\left[\begin{array}{ll}
  -1/2 & 1 \\ \\
  0 & 3/2
  \end{array}\right]\)
• -A = -1 \(\left[\begin{array}{ll}
  -1 & 2 \\ \\
  0 & 3
  \end{array}\right]\) = \(\left[\begin{array}{ll}
  1 & -2 \\ \\
  0 & -3
  \end{array}\right]\)
• 0A = 0 \(\left[\begin{array}{ll}
  -1 & 2 \\ \\
  0 & 3
  \end{array}\right]\) = \(\left[\begin{array}{ll}
  0 & 0 \\ \\
  0 & 0
  \end{array}\right]\)

Thus, matrix scalar multiplication is mathematically defined as follows:

"If A = [aᵢⱼ] ₘ ₓ ₙ and k is a scalar then kA = k [aᵢⱼ] ₘ ₓ ₙ = [kaᵢⱼ] ₘ ₓ ₙ"

i.e., the element in i^th row and j^th column of kA is obtained by multiplying the corresponding element of A by 'k'. We can visualize this in the figure below.

If A and B are matrices of the same order; and k, a, and b are scalars then:

• A and kA have the same order.
  For example, if A is a matrix of order 2 x 3 then any of its scalar multiple, say 2A, is also of order 2 x 3.
• Matrix scalar multiplication is commutative. i.e., k A = A k.
• Scalar multiplication of matrices is associative. i.e., (ab) A = a (bA).
• The distributive property works for the matrix scalar multiplication as follows:
  k (A + B) = kA + k B
  A (a + b) = Aa + Ab (or) aA + bA
• The product of any scalar and a zero matrix is the zero matrix itself. For example:
  k \(\left[\begin{array}{ll}
  0 & 0 \\ \\
  0 & 0
  \end{array}\right]\) = \(\left[\begin{array}{ll}
  0 & 0 \\ \\
  0 & 0
  \end{array}\right]\)
• The product of -1 and A gives -A which is the additive inverse of A.
  For example, the additive inverse of \(\left[\begin{array}{ll}
  -1 & 2 \\ \\
  0 & 3
  \end{array}\right]\) is (-1) \(\left[\begin{array}{ll}
  -1 & 2 \\ \\
  0 & 3
  \end{array}\right]\) = \(\left[\begin{array}{ll}
  1 & -2 \\ \\
  0 & -3
  \end{array}\right]\).

☛ Related Topics:

• Addition of Matrices
• Matrix Addition Calculator
• Types of Matrices
• Properties of Matrices

What is the Difference Between Matrix Scalar Multiplication and Matrix Multiplication?

Matrix scalar multiplication is multiplying a matrix by a scalar whereas matrix multiplication is multiplying two matrices. For any two matrices A and B, and for a scalar 'k', kA and kB represent the scalar multiplications of A and B respectively by k whereas AB represents the multiplication of matrices A and B.

How Do You Solve Matrix Scalar Multiplication?

The result of multiplying a matrix by a scalar is again a matrix of the same order where each of its elements is obtained by multiplying the corresponding elements of the original matrix by the scalar. For example, if P = \(\left[\begin{array}{ccc}
2 & -1 & 3 \\
0 & 5 & 2 \\
1 & -1 & -2
\end{array}\right]\) then

-3P

= (-3) \(\left[\begin{array}{ccc}
2 & -1 & 3 \\
0 & 5 & 2 \\
1 & -1 & -2
\end{array}\right]\)

= \(\left[\begin{array}{ccc}
-3(2) & -3(-1) & -3(3) \\
-3(0) & -3(5) & -3(2) \\
-3(1) & -3(-1) & -3(-2)
\end{array}\right]\)

= \(\left[\begin{array}{ccc}
-6 & 3 & -9 \\
0 & -15 & -6 \\
-3 & 3 & 6
\end{array}\right]\)

Can We Multiply a Matrix by a Scalar?

Yes, we can multiply a matrix by a scalar. For doing this, we just need to multiply every element of the matrix by the scalar. For example, if A = \(\left[\begin{array}{ccc}
2 & -1 & 3 \\ \\
0 & 5 & 2 \\ \end{array}\right]\) then 2A = \(\left[\begin{array}{ccc}
4 & -2 & 6\\\\
0 & 10 & 4 \\ \end{array}\right]\).

Can You Multiply a Matrix by 3?

A matrix can be multiplied by any scalar and hence it can be multiplied by 3 as well. For example, if A = \(\left[\begin{array}{ll}
1 & 2 \\
3 & 4 \\
5 & 1
\end{array}\right]\) then 3A = \(\left[\begin{array}{ll}
3 & 6 \\
9 & 12 \\
15 & 3
\end{array}\right]\).

Is Matrix Scalar Multiplication Commutative?

Yes, the matrix scalar multiplication is commutative. i.e., for any matrix M and a scalar 'a', we have aM = Ma. For example:

• 2 \(\left[\begin{array}{ll}
  1 & -1 \\
  2 & 1
  \end{array}\right]\) = \(\left[\begin{array}{ll}
  2 & -2 \\
  4 & 2
  \end{array}\right]\).
• \(\left[\begin{array}{ll}
  1 & -1 \\
  2 & 1
  \end{array}\right]\) 2 = \(\left[\begin{array}{ll}
  2 & -2 \\
  4 & 2
  \end{array}\right]\).

Can a Matrix be a Scalar?

No, a matrix cannot be a scalar. A matrix is a rectangular array of elements where the elements are arranged in rows and columns. A scalar is just a real number. Hence, a matrix cannot be a scalar.