Matrix by Scalar Calculator
Multiply or divide matrices by scalar values with detailed step-by-step solutions
Matrix Scalar Operations
Results
Result Matrix (2 × A)
Matrix Properties
Step-by-Step Solution
Example Calculations
Scalar Multiplication Example
Matrix A:
[3 4]
Scalar k = 3
3 × A:
[9 12]
Scalar Division Example
Matrix A:
[12 3]
Scalar k = 3
A ÷ 3:
[4 1]
Scalar Operation Rules
Element-wise Operation
Multiply/divide each element by scalar
C[i,j] = k × A[i,j] or A[i,j] ÷ k
Same Dimensions
Result has same size as original
Matrix dimensions unchanged
Division Restriction
Cannot divide by zero
k ≠ 0 for division operations
Mathematical Properties
Associative: (xy)A = x(yA)
Distributive: x(A + B) = xA + xB
Identity: 1 × A = A
Zero: 0 × A = Zero Matrix
Eigenvalues: λ(kA) = k × λ(A)
Determinant: det(kA) = k^n × det(A)
Understanding Matrix Scalar Operations
What is Matrix Scalar Multiplication?
Matrix scalar multiplication is the operation of multiplying every element in a matrix by a single number (scalar). Each element is multiplied individually, resulting in a matrix of the same dimensions.
How it Works
- •Element-wise operation: Each matrix element gets multiplied
- •Preserves dimensions: Result has same size as original
- •Simple calculation: C[i,j] = k × A[i,j]
Mathematical Notation
For scalar k and matrix A:
Multiplication: k × A = [k×a₁₁ k×a₁₂ ...]
Division: A ÷ k = [a₁₁/k a₁₂/k ...]
Where k ≠ 0 for division
Real-world Applications
- •Scaling transformations in computer graphics
- •Unit conversions in engineering calculations
- •Normalizing data in statistical analysis
Special Cases
Identity Scalar (k = 1)
1 × A = A (matrix unchanged)
Zero Scalar (k = 0)
0 × A = Zero Matrix (all elements become 0)
Negative Scalar (k < 0)
Changes sign of all elements
Advanced Properties
Eigenvalue Effect
If λ is eigenvalue of A, then kλ is eigenvalue of kA
Determinant Rule
det(kA) = k^n × det(A) for n×n matrix
Rank Preservation
rank(kA) = rank(A) for k ≠ 0