k·A Scalar Multiplication

Scalar Matrix Multiplication

Multiply any matrix by a scalar k. 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)