Matrix × Scalar Calculator (2×2)
Multiply every element of a 2×2 matrix by a scalar k and see the resulting matrix, its determinant, and how the scaling changes the geometric transformation visualised on the unit square.
Determinant of the scaled matrix (= k² × det A)
Original matrix A
Scalar k
Multiply each element by k
Result matrix k·A
- 1
Determinant of original matrix A
1 × 4 − 2 × 3 = -2det(A) = a₁₁·a₂₂ − a₁₂·a₂₁ - 2
k²
3 × 3 = 9 - 3
det(k·A) = k² × det(A)
9 × -2 = -18
How does this calculator work?
k · A multiplies every element of a 2×2 matrix A by the scalar k. The result matrix has elements k·aᵢⱼ, and its determinant equals k² · det(A). Enter a, b, c, d and k to get the full scaled matrix and its geometric effect on the unit square.
Formula
How this is calculated
Scalar multiplication of a matrix scales every entry by the same factor k: if A = [[a, b], [c, d]] then k·A = [[ka, kb], [kc, kd]]. No entries interact with each other — each is independently multiplied by k. This is the simplest matrix operation and follows directly from the distributive property of multiplication over addition.
The effect on the determinant is: det(k·A) = k² · det(A) for a 2×2 matrix (more generally, det(k·A) = kⁿ · det(A) for an n×n matrix). Geometrically, the determinant measures the area scaling factor of the linear transformation — multiplying by k scales every linear dimension by k, so the area (and the determinant) scales by k².
The grid diagram shows how the scaled matrix transforms the standard unit square, making the scaling intuitive.
Frequently asked questions
No. Scalar multiplication commutes: k·A = A·k. Every element is simply multiplied by k regardless of order.
(k·A)⁻¹ = (1/k) · A⁻¹, provided A is invertible and k ≠ 0. The inverse is also scaled, but by the reciprocal.
Yes. A negative scalar reflects the transformation (negative determinant), and a fractional scalar shrinks the matrix. All real k values are valid; k = 0 gives the zero matrix.
Also known as
TG we-Calculate Editorial Team. (2026). Matrix × Scalar Calculator (2×2) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/matrix-by-scalar-calculator
TG we-Calculate Editorial Team. "Matrix × Scalar Calculator (2×2)." TG we-Calculate. 2026. https://we-calculate.com/calculator/matrix-by-scalar-calculator.
TG we-Calculate Editorial Team, "Matrix × Scalar Calculator (2×2)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/matrix-by-scalar-calculator
@misc{wecalculate_matrix_by_scalar_calculator, title = {Matrix × Scalar Calculator (2×2)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/matrix-by-scalar-calculator}}, year = {2026}, note = {TG we-Calculate} }
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