Advanced Math

Matrix Calculator

Type matrices one row per line. Addition, subtraction, scalar multiplication, matrix multiplication and transpose work up to 5 × 5, with every entry's working shown. Determinants use ad − bc for 2 × 2 and cofactor expansion for 3 × 3. The 2 × 2 inverse is built from the determinant and the adjugate. Dimension mismatches and singular matrices are explained.

InputsA·B, det A, A⁻¹

OperationMatrix AOne row per line, entries separated by spaces.Matrix B

Step-by-step

waiting

Fill in the fields and press Solve. The full method will appear here, one numbered step at a time.

7 tools in Advanced Math.Back to the tool directory →

Worked example

Multiply A = [1 2; 3 4] by B = [5 6; 7 8].

1.
Matrix A is 2 × 2.
```
[ 1  2 ]
[ 3  4 ]
```
2.
Matrix B is 2 × 2.
```
[ 5  6 ]
[ 7  8 ]
```
3.
A has 2 columns and B has 2 rows, so the product exists and is 2 × 2. Each entry cᵢⱼ is row i of A times column j of B: multiply matching entries and add.

Entry by entry:

c11 = 1·5 + 2·7 = 19
c12 = 1·6 + 2·8 = 22
c21 = 3·5 + 4·7 = 43
c22 = 3·6 + 4·8 = 50

5.
Result:
```
[ 19  22 ]
[ 43  50 ]
```

Answer: [ 19 22 ] [ 43 50 ]

Note: Up to 5×5. Determinants for 2×2 and 3×3; inverses for 2×2 only.

Use “Load worked example” above to run this in the solver.

Related formulas

The formulas this tool uses, each explained with symbols, conditions and its own example in the Formula Library.

Determinant of a 2 × 2 matrixdet [[a, b], [c, d]] = ad − bc Determinant of a 3 × 3 matrixdet = a(ei − fh) − b(di − fg) + c(dh − eg)Inverse of a 2 × 2 matrix[[a, b], [c, d]]⁻¹ = (1 ÷ (ad − bc)) × [[d, −b], [−c, a]]Matrix multiplication(AB)ᵢⱼ = Σₖ Aᵢₖ × Bₖⱼ