Library / Foundations: Maths / FMA-66
Foundations: Maths
Determinant of a 2×2 matrix
Single number built from the four entries of a 2×2 matrix.
- Ref
- FMA-66
- Rev
- v1
- Field
- Foundations: Maths
- Result
- |A|
a and b are the top row of the matrix, c and d the bottom row, so the matrix is [a b; c d]. The determinant ad − bc is zero when the rows are proportional, and then the matrix has no inverse. It is also the denominator of the 2×2 inverse and of Cramer’s rule.
Single number built from the four entries of a 2×2 matrix. It is the signed area scale factor of the transformation the matrix describes. A zero determinant means the matrix has no inverse and its rows are proportional.
Calculator
- Product ad12
- Product bc2
- Inverse top-left entry, d/|A|0.4
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.
- The determinant is 10, so the matrix is invertible and a system with these coefficients has a unique solution.
- The size of the determinant is the area scale factor: the unit square maps to a parallelogram of area 10.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Entry a (row 1, column 1) | −20 to 20 | −82 | 78 | 160 | +0.8 | +32 % |
| d Entry d (row 2, column 2) | −20 to 20 | −62 | 58 | 120 | +0.8 | +24 % |
| b Entry b (row 1, column 2) | −20 to 20 | −28 | 52 | 80 | +0.8 | −16 % |
| c Entry c (row 2, column 1) | −20 to 20 | −8 | 32 | 40 | +0.8 | −8 % |
Sweep of Entry a (row 1, column 1)
| Entry a (row 1, column 1) | Determinant |
|---|---|
| −20 | −82 |
| −15 | −62 |
| −10 | −42 |
| −5 | −22 |
| 0 | −2 |
| 5 | 18 |
| 10 | 38 |
| 15 | 58 |
| 20 | 78 |
Sweep one variable
Variables and how each one contributes
Determinant
What the equation gives you. Everything else feeds it.
Entry a (row 1, column 1)
Entry a, row 1, column 1. Raises the determinant in proportion to d. Raising it by 10 % (to 3.3) moves the result by +12 %, from 10 to 11.2.
Entry b (row 1, column 2)
Entry b, row 1, column 2. Lowers the determinant in proportion to c. Raising it by 10 % (to 1.1) moves the result by −2 %, from 10 to 9.8.
Entry c (row 2, column 1)
Entry c, row 2, column 1. Lowers the determinant in proportion to b. Raising it by 10 % (to 2.2) moves the result by −2 %, from 10 to 9.8.
Entry d (row 2, column 2)
Entry d, row 2, column 2. Raises the determinant in proportion to a. Raising it by 10 % (to 4.4) moves the result by +12 %, from 10 to 11.2.
Worked example with the values above
- Main diagonal: ad = 3 × 4 = 12.
- Other diagonal: bc = 1 × 2 = 2.
- |A| = 12 − 2 = 10.
Assumptions and limits
- A 2×2 matrix of real numbers.
- Entries in any consistent units; the determinant then carries the product of the units.
Common mistakes
- Swapping the diagonals (bc − ad flips the sign).
- Treating |A| as an absolute value instead of the determinant.
- Thinking a zero determinant means a zero matrix.
Statistics and uncertainty
A determinant near zero is a warning: the matrix is nearly singular, so the solution of a system is very sensitive to errors in the coefficients, and small rounding errors can change the answer by a large factor.
Where it is used
- Testing whether two equations have a unique solution.
- Finding the inverse of a 2×2 matrix and applying Cramer’s rule.
Maths you need
- Matrices: rows and columns
- Simultaneous equations: two unknowns