Scalar Matrix Multiplication Calculator

Multiply a matrix by a scalar value with step-by-step element-wise solutions.

Enter Scalar & Matrix

Matrix A

Result

k × A =
Scalar 3
Matrix Size 2 x 2

Step-by-Step Solution

(kA)[i][j] = k × A[i][j]

What is Scalar Matrix Multiplication?

Scalar matrix multiplication is the operation of multiplying every element of a matrix by a single number, called a scalar. If A is a matrix and k is a scalar, then the result kA is a new matrix where each entry is k times the corresponding entry of A.

Unlike matrix-matrix multiplication, scalar multiplication is straightforward: simply multiply each element individually. The result always has the same dimensions as the original matrix.

How Scalar Multiplication Works

Step-by-Step Process

  1. Take the scalar value k.
  2. For each element A[i][j] in the matrix, compute k × A[i][j].
  3. Place the product in the same position of the result matrix.

Properties of Scalar Multiplication

Distributive (Scalar)

Distributes over scalar addition.

(a + b)A = aA + bA

Distributive (Matrix)

Distributes over matrix addition.

k(A + B) = kA + kB

Associative

Scalar multiplication is associative with itself.

a(bA) = (ab)A

Identity

Multiplying by 1 leaves the matrix unchanged.

1 × A = A

Applications

Scalar multiplication is used in scaling transformations in computer graphics, adjusting weights in machine learning models, normalizing matrices, and solving systems of linear equations. It is one of the most common operations performed on matrices.

Quick Tips

  • Multiplying by 0 gives the zero matrix.
  • Multiplying by -1 negates every element (additive inverse).
  • The determinant of kA for an n×n matrix is kn × det(A).
  • Scalar multiplication commutes: kA = Ak.