✖️ Matrix by Scalar Calculator
Multiply a matrix by a scalar (number)
How to Use This Calculator
Select Matrix Dimensions
Choose number of rows and columns.
Enter Scalar
Input the scalar (number) to multiply the matrix by.
Enter Matrix
Input all elements of matrix A.
View Result
See k × A where each element is multiplied by the scalar.
Formula
(kA)ᵢⱼ = k × Aᵢⱼ
Each element multiplied by the scalar
Definition:
Multiplying a matrix by a scalar k means multiplying every element by k: (kA)ᵢⱼ = k × Aᵢⱼ
Example:
k = 3, A = [1 2; 3 4]
3A = [3×1 3×2; 3×3 3×4] = [3 6; 9 12]
Properties:
- k(A + B) = kA + kB (distributive)
- (k + m)A = kA + mA (distributive)
- (km)A = k(mA) (associative)
- 1A = A (identity)
- 0A = 0 (zero matrix)
About Matrix by Scalar Calculator
The Matrix by Scalar Calculator multiplies every element of a matrix by a scalar (number). This is a fundamental operation in linear algebra, scaling the entire matrix by a constant factor. It's used in many applications including scaling transformations, gradient descent, and vector spaces.
When to Use This Calculator
- Linear Algebra: Scaling matrices
- Transformations: Scaling linear transformations
- Machine Learning: Gradient updates (learning rate × gradient)
- Physics: Scaling physical quantities
- Optimization: Step size adjustments
Why Use Our Calculator?
- ✅ Simple Operation: Multiply all elements by scalar
- ✅ Flexible Size: Supports various matrix dimensions
- ✅ Clear Display: Shows input and result
- ✅ Educational: Helps understand scalar multiplication
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Scalar: A single number (not a matrix)
- Element-wise: Every element is multiplied independently
- Scaling: Matrix is scaled uniformly in all directions
- Vector Space: Scalar multiplication is a vector space operation
- Negative Scalar: Multiplying by -1 negates all elements
Frequently Asked Questions
What is scalar multiplication?
Scalar multiplication multiplies every element of a matrix by a single number (scalar). It scales the entire matrix uniformly: (kA)ᵢⱼ = k × Aᵢⱼ.
How is this different from matrix multiplication?
Scalar multiplication is element-wise (each element × scalar). Matrix multiplication AB involves dot products of rows and columns, requiring compatible dimensions and following different rules.
Can the scalar be negative?
Yes! A negative scalar negates all elements. For example, -1 × A gives -A (the additive inverse of A).
What happens with scalar 0?
Multiplying any matrix by 0 gives the zero matrix (all elements become 0): 0A = 0.
Is scalar multiplication commutative?
Yes! kA = Ak (the scalar can be on either side). This is not true for matrix multiplication in general.