Linear Algebra • 2×2 & 3×3
Matrix Calculator
Add, subtract, multiply, find determinant, inverse, and transpose
Add
Subtract
Multiply
Determinant
Inverse
Transpose
Result
Enter matrices and click Calculate
Add, subtract, multiply, find determinant, inverse, and transpose
Enter two matrices and choose an operation — addition, subtraction, multiplication, determinant, or transpose. Results are computed instantly and shown in full matrix form, with an error message if the chosen operation isn't valid for the dimensions you've entered (for example, multiplying matrices with mismatched inner dimensions).
Addition/Subtraction: matrices must have identical dimensions; each entry is added or subtracted with the entry in the same position.
Multiplication: the number of columns in the first matrix must equal the number of rows in the second. Each entry in the result is the sum of products across a row of the first matrix and a column of the second.
Determinant: for a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc. Larger matrices use cofactor expansion.
Transpose: flips the matrix over its diagonal, turning rows into columns.
Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. If those don't match, the multiplication is undefined and the calculator will show a dimension error.
The determinant indicates whether a matrix is invertible (a zero determinant means it isn't) and, geometrically, represents the scaling factor a linear transformation applies to area or volume.
Yes — for matrices of the same size, A + B always equals B + A. Matrix multiplication, however, is generally not commutative: A×B usually does not equal B×A.