Matrix Calculator
Enter a matrix as values separated by commas and rows separated by semicolons — 4, 7; 2, 6 — then pick an operation. The answer comes with the working: the determinant is expanded along the first row into its minors, and the inverse is built as the adjugate divided by that determinant, one cofactor at a time. Determinant, inverse and transpose read matrix A alone; add, subtract and multiply use both.
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Determinant of A
10.0000
Order
2 × 2
Details
Updates as you typeDeterminant, inverse and transpose use matrix A alone.
Both matrices are read at this size, so changing it means retyping them.
Values separated by commas or spaces, rows separated by a semicolon. A plain run of four values is read as a 2 × 2.
Used by add, subtract and multiply only. The default is the identity matrix.
Summary
Determinant of A
10.0000
- Matrix A
- [4, 7; 2, 6]
- Order
- 2 × 2
- Trace (sum of the diagonal)
- 10.0000
- Determinant of Aᵀ (always equal to det A)
- 10.0000
- Invertible
- Yes — the determinant is not zero
- The determinant is the factor by which the matrix scales area (2 × 2) or volume (3 × 3). A negative determinant means the transformation also flips orientation, and a determinant of zero means it collapses space onto a line or a point — which is exactly why such a matrix has no inverse.
- Expanding along the first row is a choice, not a rule. Any row or column gives the same number, and picking one containing a zero is how the arithmetic is kept short by hand.
- Matrix B is not used by this operation, so whatever it holds is ignored.
How this is calculated
- Matrix A
- [4, 7; 2, 6]
- a₁₁ × a₂₂ = 4 × 6
- 24.0000
- a₁₂ × a₂₁ = 7 × 2
- 14.0000
- det A = a₁₁a₂₂ − a₁₂a₂₁
- 10.0000
Compare scenarios
See how one change moves the result
- CurrentYour inputs as they stand10.0000Current
Frequently asked questions
How do you find the determinant of a 3 x 3 matrix?
Expand along the first row. Each entry is multiplied by the determinant of the 2 x 2 matrix left when you delete that entry’s own row and column — its minor — and the three terms alternate in sign: det A = a11·M11 − a12·M12 + a13·M13. This calculator lists each minor and each signed term separately, so you can check the arithmetic rather than take the total on trust. Any row or column works; expanding along one that contains a zero is how the work is kept short by hand.
Why does my matrix have no inverse?
Because its determinant is zero, which makes the matrix singular. The inverse is the adjugate divided by the determinant, so a determinant of zero asks you to divide by zero and there is no answer to give. Geometrically the matrix has collapsed space onto a line or a point, and no transformation can pull it back out again. It happens whenever one row is a multiple of another, or a combination of the others — [1, 2; 2, 4] is the smallest example.
What is the adjugate, and why divide by the determinant?
The adjugate is the transpose of the cofactor matrix: replace every entry by its signed minor, then flip the result across the diagonal. The reason it works is the identity A · adj(A) = (det A) · I — multiplying a matrix by its adjugate gives the identity scaled by the determinant. Dividing through by det A therefore leaves exactly the matrix that undoes A. The page shows every cofactor, then the division, then multiplies A by the result so you can watch the identity matrix come back.
Is A times B the same as B times A?
Usually not. Matrix multiplication is not commutative, because each entry of the product is a row of the first matrix dotted with a column of the second, and swapping the operands changes which row meets which column. Some pairs do commute — anything multiplied by the identity, and a matrix with its own inverse — so this calculator works out B x A alongside A x B and says plainly whether the two agree for the pair you entered.
How do I type a matrix into a single line?
Separate the values in a row with commas or spaces, and separate rows with a semicolon: 4, 7; 2, 6 is the 2 x 2 with 4 and 7 on the top row. Newlines work too, for anything pasted from a textarea. Square brackets are ignored rather than rejected, so the calculator’s own output pastes straight back in, and a flat run of four values with no separators at all is read as a 2 x 2 — which is what a spreadsheet column becomes once a single-line input flattens it.
What does the determinant actually mean?
It is the factor by which the matrix scales area, for a 2 x 2, or volume, for a 3 x 3. A determinant of 10 means the unit square is mapped to a shape of area 10. A negative determinant means the transformation also reverses orientation, turning the plane over. A determinant of zero means the output is flat — everything is squashed onto a line or a point — which is the same fact as the matrix having no inverse.