k·A Scalar Multiplication

Scalar Matrix Multiplication

Scalar multiplication multiplies every element of a matrix by a single number k, and this calculator shows the element-by-element computation for any matrix size. Every element is multiplied by k: (kA)ᵢⱼ = k × Aᵢⱼ. Shows effect on determinant and trace.

k·A

Scalar × Matrix

Multiply all elements by constant k

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Matrix A
Quick Examples

Scalar Multiplication

For scalar k and matrix A, scalar multiplication gives a new matrix where every element is multiplied by k:

$$(kA)_{ij} = k \cdot A_{ij}$$

Properties

  • det(kA) = kⁿ det(A) for n×n matrix A
  • tr(kA) = k · tr(A)
  • k(A + B) = kA + kB (distributive)
  • (k + m)A = kA + mA
  • (km)A = k(mA)

Frequently Asked Questions

Every single element in the matrix gets multiplied by that number. The shape and structure stay the same — only the values inside change. For example, doubling every element in a matrix scales the whole thing by 2.
Yes — any real number works. A negative scalar flips the sign of every element, and a fraction like 0.5 shrinks every element proportionally. The rules are identical regardless of the scalar's value.
Yes — scalar multiplication applies element-by-element, so it works on matrices of every shape and size, from a single element up to very large matrices.

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