# Matrix by Scalar Calculator

URL: https://speedycalculator.com/calculators/math/matrix-by-scalar/
Last updated: 2026-10-30

> The matrix by scalar calculator multiplies every entry of a matrix (up to 4 × 4) by a number k.

## Inputs

- Matrix size (rows × columns)
- a₁
- b₁
- c₁
- d₁
- a₂
- b₂
- c₂
- d₂
- a₃
- b₃
- c₃
- d₃
- a₄
- b₄
- c₄
- d₄
- Scalar k

## Results

- Result of k · A

## Formula

```
(k·A)ᵢⱼ = k · aᵢⱼ
```

## Explanation

Scalar multiplication scales a matrix – converting units in a table of data, scaling a transformation or forming linear combinations of matrices.

### How to use the Matrix by Scalar Calculator

- Choose the **matrix size** – the matching grid of fields appears.
- Enter the entries **row by row** (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Each entry is multiplied by k. Multiplying by −1 gives the negative of the matrix; by 0 the zero matrix.

## Worked example

Inputs — Matrix size (rows × columns): 2 × 3; a₁: 4; b₁: 1; c₁: 2; a₂: 1; b₂: 3; c₂: 0; Scalar k: 1.5.

Results — **Result of k · A = ⎡ 6 1.5 3 ⎤
⎣ 1.5 4.5 0 ⎦**.

## Key terms

- **Scalar**: A single number.

## FAQ

**What is scalar multiplication?**
Multiplying every entry by the same number.

**Does it change the determinant?**
det(kA) = kⁿ det(A) for an n × n matrix.

**Is k·A the same as A·k?**
Yes.

**What happens with k = 0?**
You get the zero matrix.


## Sources

- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)