The formula this page uses
det [[a, b], [c, d]] = ad − bc A⁻¹ = (1 / det A) · [[d, −b], [−c, a]]
Linear Algebra Calculator
Solve linear algebra problems with clear steps, notation, and a final check. Choose a focused operation below to calculate, inspect the method, and connect the result to a visual model.
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Linear algebra
Linear Algebra Calculator explained
The short version
- A matrix is a grid of coefficients that packs a whole system of equations into one object.
- The determinant is a single number that says whether a square matrix can be undone. Zero means it cannot.
- Order matters in matrix multiplication: A times B is usually not B times A.
What each part means
SymbolWhat it means
A — Coefficient matrixThe numbers in front of the unknowns, one row per equation.
x — Unknown vectorThe column of variables you are solving for.
b — Right-hand sideThe column of constants the equations equal.
det A — DeterminantA single number. Its size is the area scale factor, and its sign says whether orientation flips.
Show your work: a full example
- The system2x + 3y = 12 and 4x − y = 10
- Write it as a matrix and a columnA = [[2, 3], [4, −1]], b = [12, 10]
- Determinant, down the diagonalsdet A = (2)(−1) − (3)(4) = −2 − 12 = −14
- Not zero, so the inverse existsA⁻¹ = (1/−14) · [[−1, −3], [−4, 2]]
- Multiply the first row of A⁻¹ by bx = ((−1)(12) + (−3)(10)) ÷ (−14) = (−12 − 30) ÷ (−14) = −42 ÷ −14 = 3
- Multiply the second row of A⁻¹ by by = ((−4)(12) + (2)(10)) ÷ (−14) = (−48 + 20) ÷ (−14) = −28 ÷ −14 = 2
- Check in both original equations2(3) + 3(2) = 6 + 6 = 12 ✓ and 4(3) − 2 = 12 − 2 = 10 ✓
A second, different case
- A different case: the determinant is zero2x + 3y = 12 and 4x + 6y = 18
- Determinant of the new coefficient matrix(2)(6) − (3)(4) = 12 − 12 = 0
- Row 2 minus 2 × Row 1, on both sides(4 − 4)x + (6 − 6)y = 18 − 24
- What is left0 = −6, which is impossible, so the system has no solution
- Change the 18 to a 24 and repeat(4 − 4)x + (6 − 6)y = 24 − 24 → 0 = 0
- Read that instead0 = 0 is always true, so now every point on the line 2x + 3y = 12 is a solution
Copy-ready example
2x + 3y = 12 and 4x − y = 10
The system
Questions about the Linear Algebra Calculator
What does a determinant of zero really mean?
It means the matrix squashes space flat: a square of area 1 gets mapped onto a line of area 0. Because you cannot un-flatten it, there is no inverse, and the system either has no solutions or infinitely many depending on b.
Why do the dimensions have to match to multiply matrices?
Each entry of the product is a row of the first matrix paired term by term with a column of the second. That pairing only works if the row and the column have the same length, which is why an m×n can only multiply an n×p.
Is the matrix the same thing as the system it came from?
It carries the same information in a shorter form. A holds only the coefficients, so you must keep b beside it. The two zero-determinant rows in the table above share the identical matrix and differ only in b, yet one has no solutions and the other has infinitely many.
Do I need the inverse to solve a system?
No, and for anything bigger than 2×2 you usually should not. Row-reducing the augmented matrix takes fewer operations and keeps working when the determinant turns out to be zero, whereas the inverse method simply breaks.
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