Matrix Scalar Multiplication Calculator
Multiply every element of a matrix by a scalar constant k. This scales the entire matrix uniformly.
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Notes
Scalar Multiplication of a Matrix
Multiplying a matrix by a scalar k scales every element by k. This is the most fundamental matrix operation.
Example: Multiply by 3
Original A:
| 1 | 2 |
| 3 | 4 |
3A:
| 3 | 6 |
| 9 | 12 |
The determinant of kA equals k^n · det(A) for an n×n matrix A.
See also
- Matrix Calculations Guide — Scalar Operations — In-depth notes on Scalar Operations with worked examples
Frequently Asked Questions
What happens to the determinant when you scale a matrix?
For an n×n matrix, det(kA) = k^n · det(A). For example, multiplying a 2×2 matrix by 3 multiplies its determinant by 9.
Does scalar multiplication change the rank?
No (unless k = 0). Multiplying by a nonzero scalar preserves rank because it is equivalent to a sequence of row operations.
What happens when k = 0?
Multiplying by zero gives the zero matrix, which has rank 0 (regardless of the original matrix's rank).