About this calculator
A matrix is a grid of numbers. Type yours with spaces between the entries and a semicolon between rows, so a 2×2 matrix is 1 2; 3 4.
Choose what to do: find the determinant, the inverse, the transpose, or add or multiply two matrices. The result is shown row by row, with the working beneath.
Worked examples
Real numbers, worked out by the same calculator. Press “Use these numbers” to try one above.
Determinant of [1 2; 3 4]
- Determinant
- -2
- Invertible?
- Yes
The determinant is -2. It is not zero, so the matrix has an inverse.
Show the working
- 2×2 determinant = ad − bc = 1 × 4 − 2 × 3 = -2.
Inverse of [1 2; 3 4]
- Inverse row 1
- [ -2, 1 ]
- Inverse row 2
- [ 1.5, -0.5 ]
- Determinant
- -2
The inverse is below. Multiplying the matrix by it gives the identity matrix.
Show the working
- 2×2 determinant = ad − bc = 1 × 4 − 2 × 3 = -2.
- Because the determinant is not zero, row-reduce [A | I] until the left side is the identity; the right side is then the inverse.
Multiply two 2×2 matrices
- Product row 1
- [ 19, 22 ]
- Product row 2
- [ 43, 50 ]
A × B is a 2×2 matrix.
Show the working
- Each entry is a row of A times a column of B: (AB)ij = Σ Aik × Bkj.
- For example, the top-left entry is 1 × 5 + 2 × 7 = 19.
- Row 1: 19, 22
- Row 2: 43, 50
Transpose of a 2×3 matrix
- Transpose row 1
- [ 1, 4 ]
- Transpose row 2
- [ 2, 5 ]
- Transpose row 3
- [ 3, 6 ]
The transpose of this 2×3 matrix is 3×2.
Show the working
- Swap rows and columns: entry (i, j) moves to (j, i).
- Row 1: 1, 4
- Row 2: 2, 5
- Row 3: 3, 6
How each operation works
- Determinant: for 2×2 it is ad − bc; larger ones use row elimination
- Inverse: the matrix that multiplies with A to give the identity; it exists only when the determinant is not zero
- Transpose: swap rows and columns
- Multiply: each entry is a row of A times a column of B; needs A's columns to equal B's rows
- Add: add matching entries; the sizes must match
What the determinant tells you
A determinant of zero means the matrix is singular: it has no inverse, and a system of equations built from it either has no solution or infinitely many. A non-zero determinant means a single unique solution exists. For a 2×2 matrix the determinant is also the area scale factor of the transformation.
Order matters
Matrix multiplication is not commutative: A × B is usually different from B × A. Try the example both ways round. Addition does not have this problem.
Frequently asked questions
How do I enter a matrix?
Put spaces between entries and a semicolon between rows: 1 2 3; 4 5 6 is a 2×3 matrix.
How big can the matrices be?
Up to 6 rows and 6 columns.
Why can't it find the inverse?
Because the determinant is zero (a singular matrix) or the matrix is not square.
Can I use decimals and negatives?
Yes: for example -1.5 0; 2 3.25.
Can I multiply a 2×3 by a 3×2?
Yes. The inner sizes (3 and 3) match, and the result is 2×2.
Formulas tested against hand-worked answers. Last reviewed 29 September 2026. These calculators do arithmetic only; they are not financial, tax or legal advice.