Math
Scalar Matrix Multiplication Calculator
Enter a scalar and a matrix to multiply every entry by that number.
The calculator keeps fractions exact, lists each k·a_ij step, and for square matrices shows how the determinant scales by kⁿ while the trace scales by k.
Any number, decimal, or fraction. Negative values are fine.
One row per line. Separate entries with spaces or commas. Integers, decimals, and fractions like 1/3 all work. Up to 8 x 8.
Try an example
Result
kA
—
- Scalar
- —
- Size
- —
- Determinant scale factor
- —
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Study path
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What scalar multiplication does
Multiplying a A rectangular grid of numbers arranged in rows and columns. by a scalar k multiplies every single entry by k. The matrix keeps its shape. This is different from matrix multiplication, which combines rows and columns and can change the shape.
Properties
| Property | Formula |
|---|---|
| Distributes over matrix addition | k(A + B) = kA + kB |
| Distributes over scalar addition | (k + m)A = kA + mA |
| Associative with scalars | k(mA) = (km)A |
| Identity scalar | 1·A = A |
| Zero scalar | 0·A = 0 (the zero matrix) |
| With products | k(AB) = (kA)B = A(kB) |
How to use it
- Type the scalar (for example 3, -0.5, or 2/3).
- Enter the matrix, one row per line.
- Read kA and open Show the work for every entry.
Common mistakes
- Multiplying only one row or the diagonal. Every entry gets multiplied.
- Writing det(kA) = k·det(A). The correct rule is kⁿ·det(A), because k multiplies each of the n rows.
- Confusing kA with A^k. A^k means multiplying A by itself k times.
How to type a matrix into this calculator
- Put each row on its own line. Separate the entries in a row with spaces or commas: 2 1 1 or 2, 1, 1.
- Fractions such as 1/3 and decimals such as 0.25 are accepted and kept exact, so 1/3 stays 1/3 instead of becoming 0.3333.
- Every row must have the same number of entries. A row with a missing entry produces a clear error instead of a wrong answer.
- The largest size supported is 8 x 8. Brackets and semicolons are ignored, so [1 2; 3 4] works too.
Worked examples
Integer scalar
3 × [1 -2 3; 4 0 -5]
kA
[3, -6, 9; 12, 0, -15]
Fraction scalar
1/2 × [4 6; -8 3]
kA
[2, 3; -4, 3/2]
Negative scalar
-1 × A flips every sign
kA
[-1, 2; -3, -4]
Zero scalar
0 × A is the zero matrix
kA
[0, 0; 0, 0]
Decimal scalar and determinant scaling
2.5 × [1 2; 3 4]: det goes from -2 to 2.5² × (-2) = -12.5
kA
[5/2, 5; 15/2, 10]
Invalid scalar
Letters are rejected
kA
Error
Frequently asked questions
How do you multiply a matrix by a scalar?+
Multiply every entry by the scalar. For 3 × [1 2; 3 4] you get [3 6; 9 12].
Does scalar multiplication change the size of the matrix?+
No. An m x n matrix stays m x n. Only the entries change.
What happens to the determinant when you multiply a matrix by k?+
For an n x n matrix, det(kA) = kⁿ·det(A). Doubling a 3 x 3 matrix multiplies its determinant by 8.
Can I multiply a matrix by a fraction or decimal?+
Yes. Every entry is multiplied by the fraction, and this calculator keeps the result exact.
Is scalar multiplication the same as matrix multiplication?+
No. Scalar multiplication scales entries individually and always works. Matrix multiplication combines rows with columns and has a dimension rule.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026