# Matrix by Scalar Calculator

Created by Anna Szczepanek, PhD
Reviewed by Rijk de Wet
Last updated: Sep 16, 2022

If you've ever wondered how to multiply a matrix by a number, this matrix by scalar calculator is a perfect tool for you. Here, you'll discover all there is about multiplying matrices by scalars. As a bonus, we'll tell you how to divide a matrix by a number.

If you're familiar with some more advanced concepts in linear algebra, you'll be able to find here the answers to the following questions:

• What is the determinant of a matrix multiplied by a scalar?
• What are the eigenvalues of a matrix multiplied by a scalar?

💡 This tool covers the multiplication of a matrix by a scalar. If you're interested in multiplying a matrix by another matrix, visit:

## How do I multiply a matrix by a scalar?

Multiplying a matrix by a scalar (i.e., by a single number) is pretty straightforward: you just need to multiply each element of your matrix by this scalar. The result will be a matrix of the same size as your initial matrix.

To see how the multiplication of a matrix by scalar works in practice, let's use Omni's matrix by scalar calculator.

## How to use this matrix by scalar calculator?

To use this tool, you need to specify two input entities: the scalar and the matrix that you want to multiply together. For the matrix, you'll need to first choose its size and then input its elements. If an element is zero, then you might as well leave that field blank: our matrix by scalar calculator will know how to handle them.

Now that we know how to multiply a matrix by a scalar, it's time to discuss what properties this mathematical operation has.

## Properties of matrix by scalar multiplication

Let A and B be two matrices of the same size, and let x and y be numbers. The matrix by scalar multiplication operation has the following properties:

• Matrix by scalar multiplication is associative:
(xy)A = x(yA);
x(A + B) = xA + xB; and
• The neutral element is 1, meaning that
1A = A.

## How do I divide a matrix by a number?

Technically, to divide a matrix by a scalar, you need to multiply this matrix by the reciprocal of this scalar. In practice, you divide each coefficient of your matrix by the scalar and that's it. Remember that only non-zero scalars are suitable for division!

## FAQ

### What are the eigenvalues of a matrix multiplied by a scalar?

If you multiply a square matrix by a scalar, then each of its eigenvalues will get multiplied by the same scalar. That is, if λ is an eigenvalue of A associated with the eigenvector v, then kλ is an eigenvalue of kA associated with the same eigenvector v, where k is a scalar.

### What is the determinant of a matrix multiplied by a scalar?

If A is a square matrix of size n and k is a scalar, then det(kA)=kⁿ det(A).

### What is a matrix multiplied by a zero?

If you multiply any matrix by 0, you obtain a matrix where every element is equal to 0. This matrix is often called a zero matrix.

### What is the identity matrix multiplied by a number ?

To multiply the identity matrix by a number (a scalar), you need to:

1. Recall that the identity matrix has 1s on its diagonal and 0s elsewhere.
2. To multiply the identity matrix by a scalar k, you need to multiply each matrix coefficient by k.
3. Write down each product into the respective field in the resulting matrix.
4. The result you obtain is the matrix that has k on its diagonal and 0 elsewhere.
Anna Szczepanek, PhD
Number
Enter the number by which the matrix will be multiplied.
k
Matrix
Choose matrix size and enter the coeffients into the appropriate fields. Blanks are interpreted as zeros.
Number of rows
2
Number of columns
2
A=
 ⌈ a1 a2 ⌉ ⌊ b1 b2 ⌋
a₁
a₂
b₁
b₂
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