Matrix Scalar Multiplication Calculator

Math Calcs Scalar Multiplication Calculator is a online tool for calculating the resultant matrix after the multiplication of input matrix with a scalar value.

Scalar Multiplication is not a matrix multiplication. Matrix multiplication is a different operation.

Related Calculator

• Matrix Multiplication Calculator

How to use Matrix Scalar Multiplication Calculator?

To use the Scalar Multiplication Calculator above:

• There are two input boxes for entering the matrix dimensions (rows and columns)
• After you enter the dimension values, the matrix element input boxes will be rendered in the screen.
• Enter the value of each elements for matrix in each box.
• There is another input box for scalar value. Enter the value you want to multiply the matrix with in that input box.
• Click on Calculate.

There is a placeholder in each matrix input element box. For example first box have placeholder of "A11". That is a index of element, for example A11 is first row and first column element of matrix A.

Similarly A12 is first row and second column element of matrix A.

How to do Scalar Matrix Multiplication?

To perform matrix scalar multiplication for matrix M with scalar value S. The resultant matrix P will be:

\(M = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\)

\(S = 5
\)

\(P = \begin{bmatrix}
5 \cdot a & 5 \cdot b \\
5 \cdot c & 5 \cdot d
\end{bmatrix}

\)

Scalar Multiplication Properties of Matrix

Let S be a scalar value (S1, S2) and A, B be matrices. Scalar Multiplication Properties are as follows:

• Distributive over matrix addition. That is, S * (A + B) = S * A + S * B
• Associative. That is, S1 * (S2 * A) = (S1 * S2) * A
• Distributive over scalar addition. That is, (S1 + S2) * A = S1 * A + S2 * A
• Commutative. That is, S * A = A * S
• Distributive over matrix subtraction. That is, S * (A - B) = S * A - S * B

FAQs

Is Scalar Matrix Multiplication Commutative?

Yes scalar matrix multiplication is commutative. That is, S * A = A * S

\(S = 5
A = \begin{bmatrix}
2 & 3 \\
4 & 5
\end{bmatrix}
\)

\(S \cdot A = \begin{bmatrix}
5 \cdot 2 & 5 \cdot 3 \\
5 \cdot 4 & 5 \cdot 5
\end{bmatrix}
= \begin{bmatrix}
10 & 15 \\
20 & 25
\end{bmatrix}
\)

\(A \cdot S = \begin{bmatrix}
2 \cdot 5 & 3 \cdot 5 \\
4 \cdot 5 & 5 \cdot 5
\end{bmatrix}
= \begin{bmatrix}
10 & 15 \\
20 & 25
\end{bmatrix}
\)

How does multiplying matrix by scalar affect the determinant?

When a inversible matrix M is multiplied by a scalar S, the determinant of the matrix M will also be multiplied by the scalar S to the power of matrix size.

\(M = \begin{bmatrix}
2 & 3 \\
4 & 5
\end{bmatrix}
\)

\(det(M) = 2 \times 5 - 3 \times 4 = 10 - 12 = -2
\)

\(S = 5
\)

\(S \cdot M = \begin{bmatrix}
5 \cdot 2 & 5 \cdot 3 \\
5 \cdot 4 & 5 \cdot 5
\end{bmatrix}
= \begin{bmatrix}
10 & 15 \\
20 & 25
\end{bmatrix}
\)

\(det(S \cdot M) = 10 \times 25 - 15 \times 20 = 250 - 300 = -50

\)

The scalar will be raised to the power of matrix size and multiplied with the determinant of the matrix.

\(
S^2 \times det(M) = 5^2 \times -2 = 25 \times -2 = -50
\)

Does scalar matrix multiplication affects the matrix size?

No, scalar matrix multiplication does not changes matrix sizes, it only scales the elements of the matrix.

Scalar Matrix Multiplication Examples

Example 1

Let's multiply the following 4 x 1 matrix with scalar value 41

\(O = \begin{bmatrix}
4.1 \\
3.2 \\
2.3 \\
1.4

\end{bmatrix}
\)

\(41 \cdot O = \begin{bmatrix}
41 \cdot 4.1 \\
41 \cdot 3.2 \\
41 \cdot 2.3 \\
41 \cdot 1.4
\end{bmatrix}

= \begin{bmatrix}

168.1 \\
131.2 \\
94.3 \\
57.4

\end{bmatrix}
\)