Library / Foundations: Maths / FMA-68

Foundations: Maths

Determinant of a scalar multiple of a matrix

Multiplying every entry of an n×n matrix by k multiplies every row by k, so the determinant is multiplied by k to the power n, not by k.

Ref: FMA-68
Rev: v1
Field: Foundations: Maths
Result: |kA|

| k A | = k^{n} | A | (A is n \times n)

For an n×n matrix, multiplying every entry by k multiplies each of the n rows by k, and each row multiplication scales the determinant by k. So |kA| = kⁿ|A|, not k|A|. For a 3×3 matrix, doubling every entry multiplies the determinant by 8.

Multiplying every entry of an n×n matrix by k multiplies every row by k, so the determinant is multiplied by k to the power n, not by k. Doubling a 3×3 matrix multiplies its determinant by 8.

Calculator

Solve for

|kA|Determinant of kAComputed

kScalar multiplier

−1010

nOrder of the square matrix

110

|A|Determinant of the original matrix

−100100

Scale factor kⁿ8
Factor compared with the naive guess k4

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Determinant of kA () current value 40

kScalar multiplierKey driver

−5,0005,000

nOrder of the square matrix

105,120

|A|Determinant of the original matrix

−800800

Scaling a 3×3 matrix by 2 scales its determinant by 8, not by 2.
With a negative k and an odd n the sign of the determinant flips.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
k Scalar multiplier	−10 to 10	−5,000	5,000	10,000	+0.4	+72.8 %
n Order of the square matrix	1 to 10	10	5,120	5,110	+1	+100 %
\|A\| Determinant of the original matrix	−100 to 100	−800	800	1,600	+4	+80 %

Sweep of Scalar multiplier

Scalar multiplier	Determinant of kA
−10	−5,000
−7.5	−2,109
−5	−625
−2.5	−78.13
0	0
2.5	78.13
5	625
7.5	2,109
10	5,000

Sweep one variable

Vary

Determinant of kA ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

|kA|

Determinant of kA

Result ·

What the equation gives you. Everything else feeds it.

Scalar multiplier

· typical −10 to 10

The scalar multiplying every entry. Raises the determinant by the power n, so the percentage effect is n times larger. Raising it by 10 % (to 2.2) moves the result by +33.1 %, from 40 to 53.24.

Order of the square matrix

· typical 1 to 10

The order of the square matrix (number of rows). Changes the scale factor kⁿ; each extra row multiplies the determinant by another factor of k. Raising it by 10 % (to 3.3) moves the result by +0 %, from 40 to 40.

|A|

Determinant of the original matrix

· typical −100 to 100

The determinant of the original matrix. Raises the result in direct proportion. Raising it by 10 % (to 5.5) moves the result by +10 %, from 40 to 44.

Worked example with the values above

Scale factor: kⁿ = 2^3 = 8.
|kA| = 8 × 5.
|kA| = 40.

Assumptions and limits

A square matrix of order n and a real scalar k.
The determinant of the original matrix is known and finite.

Common mistakes

Using k|A| in place of kⁿ|A|.
Using the wrong order n (the number of rows, not the number of entries).
Forgetting that kⁿ is negative for negative k and odd n.

Statistics and uncertainty

Because the scale factor is kⁿ, a 1 % error in k gives an n % error in the determinant.

Where it is used

Scaling resistance, conductance or stiffness matrices and checking the effect on the determinant.
Sanity-checking determinant calculations.

Maths you need

Determinants: effect of row operations
Powers: kⁿ

Related formulas

Practise this formula Open the matrix calculator