Matrix calculator
A 2×2 matrix [[a, b], [c, d]] has determinant ad − bc, and it is invertible exactly when that is not zero. A zero determinant means the rows are multiples of each other and the system either has no solution or infinitely many — never one. Every minor and cofactor is printed, because the answer is rarely what you came to check.
Determinant, inverse and the solution of a system — with every minor and cofactor printed, because the answer is not what you came to check.
Inverses checked by multiplying back to the identity · How we check
Matrix
Right-hand side
Determinant
-54
Non-zero, so the matrix is invertible and the system has exactly one solution.
Minors and cofactors
Each minor is the determinant of what is left when you cross out that cell's row and column. The cofactor is the minor with a sign from the checkerboard, and the determinant is any row multiplied by its cofactors and added.
| Cell | Value | Minor | Cofactor |
|---|---|---|---|
| 1,1 | 2 | -26 | -26 |
| 1,2 | -1 | -2 | 2 |
| 1,3 | 0 | 5 | 5 |
| 2,1 | 1 | 2 | -2 |
| 2,2 | 3 | -4 | -4 |
| 2,3 | 4 | 10 | -10 |
| 3,1 | 0 | -4 | -4 |
| 3,2 | 5 | 8 | -8 |
| 3,3 | -2 | 7 | 7 |
Inverse
The adjugate — the cofactor matrix transposed — divided by the determinant of -54. That division is the whole reason a singular matrix has no inverse.
The system solved, by Cramer's rule
| x1 | 0.7778 | -42 ÷ -54 |
| x2 | 0.5556 | -30 ÷ -54 |
| x3 | -0.1111 | 6 ÷ -54 |
Each unknown is the determinant of the matrix with its column replaced by the right-hand side, divided by the determinant of the matrix itself.
What a determinant is actually measuring
A matrix is a transformation, and its determinant is how much that transformation scales area or volume. The unit square becomes a parallelogram; the determinant is its area, with a sign that records whether the orientation flipped. Everything else follows: a determinant of zero means the square was flattened, which is why the transformation cannot be reversed and why the system it represents has no unique solution.
That reading also explains why determinants multiply. Apply one transformation and then another and the areas scale twice, so the determinant of a product is the product of the determinants — a fact that is tedious to prove by arithmetic and obvious geometrically.
Questions people actually ask
- What does the determinant mean?
- The factor by which the matrix scales area in two dimensions or volume in three. A determinant of 3 triples areas; a determinant of −1 preserves them and flips orientation; a determinant of zero collapses everything onto a line or a point, which is why nothing can be inverted from there — the information is gone.
- How do you find a 3×3 determinant by hand?
- Expand along a row: multiply each entry by the determinant of the 2×2 left when you cross out its row and column, alternate the signs, and add. The table above prints those minors so you can check yours one at a time rather than only comparing a final answer.
- Why does a zero determinant mean no inverse?
- Because the inverse is the adjugate divided by the determinant, and dividing by zero has no value. The geometric reading is the same fact: a transformation that squashes space onto a lower dimension cannot be undone, because many different starting points ended up in the same place.
- What is Cramer’s rule?
- Each unknown equals the determinant of the matrix with that unknown’s column replaced by the right-hand side, divided by the determinant of the matrix. It is elegant for two and three unknowns and hopeless beyond that — the work grows factorially, which is why real software uses elimination instead.
- Why only 2×2 and 3×3?
- Because those are the sizes worked by hand, and the methods shown here — cofactor expansion and Cramer’s rule — are the hand methods. A 6×6 determinant by cofactors is 720 terms. Past three, the honest answer is Gaussian elimination in software rather than a larger grid on a web page.