Scaling a Matrix by a Constant Factor

Easy

In linear algebra, scalar multiplication of a matrix is a fundamental operation where every element of a matrix is multiplied by a single constant value (called a scalar). This operation uniformly scales all entries of the matrix, effectively stretching or shrinking the values by the same factor. Given a matrix A of dimensions n × m (where n is the number of rows and m is the number of columns) and a scalar k, the scaled matrix B is computed such that each element Bᵢⱼ is: $$B_{ij} = k \cdot A_{ij}$$ This operation preserves the structure and dimensions of the original matrix while transforming its values proportionally. Key Properties of Scalar Multiplication: Dimension Preservation: The resulting matrix has the exact same dimensions as the original matrix Distributive Property: k(A + B) = kA + kB for matrices A, B of the same dimensions Associative with Scalars: (ab)M = a(bM) for scalars a, b and matrix M Identity Element: Multiplying by 1 leaves the matrix unchanged Zero Scalar: Multiplying by 0 produces a zero matrix Your Task: Write a Python function that takes a matrix and a scalar value, then returns a new matrix where every element has been multiplied by the scalar. The function should handle matrices of any valid dimensions and work correctly with both positive and negative values, including integers and floating-point numbers.

Example 1

Input

matrix = [[1, 2], [3, 4]]
scalar = 2

Output

[[2, 4], [6, 8]]

Explanation

Each element of the 2×2 matrix is multiplied by the scalar 2: • Position (0,0): 1 × 2 = 2 • Position (0,1): 2 × 2 = 4 • Position (1,0): 3 × 2 = 6 • Position (1,1): 4 × 2 = 8 The resulting scaled matrix is [[2, 4], [6, 8]].

Example 2

Input

matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
scalar = 3

Output

[[3, 6, 9], [12, 15, 18], [21, 24, 27]]

Explanation

Each element of the 3×3 matrix is multiplied by the scalar 3: • Row 0: [1×3, 2×3, 3×3] = [3, 6, 9] • Row 1: [4×3, 5×3, 6×3] = [12, 15, 18] • Row 2: [7×3, 8×3, 9×3] = [21, 24, 27] The resulting scaled matrix maintains the 3×3 structure with all values tripled.

Example 3

Input

matrix = [[-1, 2], [3, -4]]
scalar = 5

Output

[[-5, 10], [15, -20]]

Explanation

Scalar multiplication works correctly with negative values, preserving their sign: • Position (0,0): -1 × 5 = -5 (negative remains negative) • Position (0,1): 2 × 5 = 10 (positive remains positive) • Position (1,0): 3 × 5 = 15 (positive remains positive) • Position (1,1): -4 × 5 = -20 (negative remains negative) The scaling operation uniformly magnifies all values while maintaining their original signs.

Constraints

1 ≤ n, m ≤ 100 (matrix dimensions)
-10⁶ ≤ matrix[i][j] ≤ 10⁶
-10⁶ ≤ scalar ≤ 10⁶
The matrix will always have at least one row and one column
All rows of the matrix will have the same length (valid matrix format)
The result should maintain the exact data type consistency with the inputs