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Foundations: Maths
Determinant of a 3×3 matrix
Determinant of a 3×3 matrix with rows (a b c), (d e f) and (g h i), found by expanding along the first row with signed 2×2 minors.
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- FMA-67
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- Foundations: Maths
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- |A|
The matrix has rows (a b c), (d e f) and (g h i). The first-row expansion adds and subtracts three 2×2 determinants (the minors) with signs +, −, +. A zero value means the matrix is singular. The determinant changes sign if two rows are swapped and is unchanged if a multiple of one row is added to another.
Determinant of a 3×3 matrix with rows (a b c), (d e f) and (g h i), found by expanding along the first row with signed 2×2 minors. A zero value means the matrix is singular: it has no inverse and the system it describes has no unique solution.
Calculator
- Minor of a, ei − fh3
- Minor of b, di − fg−2
- Minor of c, dh − eg1
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 4, so a system with these coefficients has a unique solution and the matrix has an inverse.
- For three mesh currents, a resistance matrix of independent meshes has a positive determinant. The matrix calculator solves the whole system and shows the working.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| e Entry e (row 2, column 2) | −20 to 20 | −84 | 76 | 160 | +0.8 | +80 % |
| a Entry a (row 1, column 1) | −20 to 20 | −62 | 58 | 120 | +0.8 | +60 % |
| i Entry i (row 3, column 3) | −20 to 20 | −62 | 58 | 120 | +0.8 | +60 % |
| b Entry b (row 1, column 2) | −20 to 20 | −34 | 46 | 80 | +0.8 | +40 % |
| d Entry d (row 2, column 1) | −20 to 20 | −34 | 46 | 80 | +0.8 | +40 % |
| f Entry f (row 2, column 3) | −20 to 20 | −34 | 46 | 80 | +0.8 | +40 % |
| h Entry h (row 3, column 2) | −20 to 20 | −34 | 46 | 80 | +0.8 | +40 % |
| c Entry c (row 1, column 3) | −20 to 20 | −16 | 24 | 40 | +0.8 | +20 % |
| g Entry g (row 3, column 1) | −20 to 20 | −16 | 24 | 40 | +0.8 | +20 % |
Sweep of Entry e (row 2, column 2)
| Entry e (row 2, column 2) | Determinant |
|---|---|
| −20 | −84 |
| −15 | −64 |
| −10 | −44 |
| −5 | −24 |
| 0 | −4 |
| 5 | 16 |
| 10 | 36 |
| 15 | 56 |
| 20 | 76 |
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 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to 2.2) moves the result by +15 %, from 4 to 4.6.
Entry b (row 1, column 2)
Entry b, row 1, column 2 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to −1.1) moves the result by −5 %, from 4 to 3.8.
Entry c (row 1, column 3)
Entry c, row 1, column 3 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to 0) moves the result by +0 %, from 4 to 4.
Entry d (row 2, column 1)
Entry d, row 2, column 1 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to −1.1) moves the result by −5 %, from 4 to 3.8.
Entry e (row 2, column 2)
Entry e, row 2, column 2 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to 2.2) moves the result by +20 %, from 4 to 4.8.
Entry f (row 2, column 3)
Entry f, row 2, column 3 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to −1.1) moves the result by −5 %, from 4 to 3.8.
Entry g (row 3, column 1)
Entry g, row 3, column 1 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to 0) moves the result by +0 %, from 4 to 4.
Entry h (row 3, column 2)
Entry h, row 3, column 2 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to −1.1) moves the result by −5 %, from 4 to 3.8.
Entry i (row 3, column 3)
Entry i, row 3, column 3 of the matrix. Changes the determinant in proportion to its cofactor, the signed 2×2 determinant of the entries that are not in its row or column. Raising it by 10 % (to 2.2) moves the result by +15 %, from 4 to 4.6.
Worked example with the values above
- Minors: ei − fh = 3, di − fg = −2, dh − eg = 1.
- Expand: 2(3) − −1(−2) + 0(1).
- |A| = 4.
Assumptions and limits
- A 3×3 matrix of real numbers.
- Entries in any consistent units; the determinant then carries the product of three units.
Common mistakes
- Forgetting the minus sign on the middle term of the expansion.
- Mixing up row and column order when typing the entries.
- Using the 2×2 shortcut ad − bc, which does not apply to 3×3 matrices.
Statistics and uncertainty
Determinants of nearly singular matrices are small differences of large products, so rounding and measurement errors have a large relative effect. Compare the determinant with the size of the products before trusting a solution.
Where it is used
- Checking that a three-equation system has a unique solution.
- Cramer’s rule for three unknowns, such as three mesh currents.
Maths you need
- 2×2 determinants: ad − bc
- Cofactor expansion: signs +, −, +