Matrix Scalar Multiplication Calculator
Scalar multiplication is the simplest matrix operation: every entry is multiplied by the same number. It scales the whole matrix uniformly without changing its dimensions, and it underlies linear combinations, vector spaces, and the distributive laws of linear algebra. This calculator takes up to three rows of comma-separated numbers and a scalar, then returns each scaled row. It also reports the sum of all scaled entries as a quick check. Enter rows of any equal length; the operation works element by element.
Scalar multiplication rule
(kA)[i][j] = k * A[i][j]
Every entry scaled by the same k
Dimensions unchanged
det(kA) = k^n * det(A) for n x n
kA = Ak (commutative)
Each element of the matrix is multiplied by the scalar k. The shape of the matrix is preserved and the operation distributes over matrix addition.
Notes on scalar multiplication
- Leave a row blank to skip it; rows are processed independently.
- Each entry in a row is multiplied by the same scalar.
- Scaling a square matrix by k multiplies its determinant by k to the matrix size power.
- Multiplying by one leaves the matrix unchanged; multiplying by zero gives the zero matrix.
- The sum output adds all scaled entries as a sanity check.
Scalar multiplication: frequently asked questions
What is scalar multiplication of a matrix?
Scalar multiplication multiplies every entry of a matrix by the same number, called the scalar. The result is a matrix of the same dimensions where each element is the original element times the scalar.
What is the rule for scalar multiplication?
If k is the scalar and A is the matrix, then (kA) at position (i,j) equals k times A at (i,j). Every entry is scaled identically, so the shape of the matrix does not change.
How does scalar multiplication affect the determinant?
For an n by n matrix, multiplying the whole matrix by a scalar k multiplies the determinant by k to the power n. For a 3x3 matrix, scaling by k multiplies the determinant by k cubed.
Is scalar multiplication commutative?
Yes. Multiplying a matrix by a scalar gives the same result whether the scalar is written on the left or the right: kA equals Ak. Scalar multiplication is also distributive over matrix addition.
What does multiplying by zero or one do?
Multiplying a matrix by the scalar one leaves it unchanged. Multiplying by the scalar zero produces the zero matrix, where every entry is zero, regardless of the original values.
Official sources
- NIST Digital Library of Mathematical Functions: Determinants and linear operators.
- NIST/SEMATECH e-Handbook of Statistical Methods: Linear algebra reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.