Scalar Multiplication
Multiply every entry of a matrix by a scalar.
Formula
Variables
Example
3·[[1,2],[3,4]] = [[3,6],[9,12]].
How it works
Scalar multiplication means every entry of a matrix is scaled by the same number c. The rule (cA)_{ij} = c a_{ij} says that the entry in row i, column j of the new matrix is just c times the corresponding entry of the original matrix, so the matrix keeps the same shape while its size changes. This works because a matrix can be viewed as a table of numbers, and multiplying by a scalar applies the same scaling to each component individually.
When to use it
Use scalar multiplication when you need to enlarge, shrink, or negate a matrix in a linear algebra problem. It appears in matrix algebra, solving systems, transformations, and any situation where every component of a data table or vector must be scaled equally.
Worked example, step by step
Let A = [[2, -1, 5], [0, 4, 3]] and c = -2. Compute -2A by multiplying each entry of A by -2. First row: -2·2 = -4, -2·(-1) = 2, -2·5 = -10. Second row: -2·0 = 0, -2·4 = -8, -2·3 = -6. So, -2A = [[-4, 2, -10], [0, -8, -6]].
Common mistakes to avoid
- Multiplying only the first row or first column instead of every entry
- Forgetting that a negative scalar changes the sign of each entry
- Changing the matrix dimensions instead of keeping the same shape
Did You Know?
Multiplying a matrix by a scalar simply scales every entry — stretching the whole transformation.
Share this formula
More in Linear Algebra
View allDeterminant (2×2)
IntermediateDeterminant of a 2×2 matrix.
Dot Product
IntermediateScalar product of two vectors.
Cross Product Magnitude
AdvancedMagnitude of the vector perpendicular to two vectors.
Vector Magnitude
IntermediateLength of a vector.
Eigenvalue Equation
AdvancedDefines eigenvalues and eigenvectors of a matrix.
Inverse of a 2×2 Matrix
AdvancedInverse of a 2×2 matrix (ad−bc ≠ 0).