Matrix by Scalar Calculator
Multiply a matrix by a scalar value.
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How Matrix by Scalar Multiplication Works
When multiplying a matrix by a scalar, each element of the matrix is multiplied by the scalar value:
\[ (cA)_{ij} = c \cdot A_{ij} \]
Example for a 2×2 matrix:
\[ c \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} c \cdot a & c \cdot b \\ c \cdot c & c \cdot d \end{pmatrix} \]
Properties:
- Commutative: cA = Ac
- Associative: (cd)A = c(dA)
- Distributive over matrix addition: c(A + B) = cA + cB
- Distributive over scalar addition: (c + d)A = cA + dA
- Multiplicative identity: 1A = A
Applications:
- Scaling transformations in computer graphics
- Resizing images
- Adjusting weights in neural networks
- Scaling physical quantities in physics
- Economic scaling factors
Visual Interpretation:
- Positive scalar: Stretches or shrinks the matrix
- Negative scalar: Reflects and stretches/shrinks
- Scalar > 1: Expansion
- 0 < scalar < 1: Contraction
- scalar = 0: Zero matrix