Matrix Calculator
Perform matrix operations online: add, subtract, multiply matrices, calculate determinant, transpose and inverse. Supports 2x2 and 3x3 matrices with step-by-step solutions.
How to Use
- Choose the matrix size: 2×2 or 3×3.
- Select the desired operation (Add, Subtract, Multiply, Determinant, Transpose, or Inverse).
- Enter values into Matrix A (and Matrix B if required by the operation).
- Click Calculate to see the result and a step-by-step explanation.
Formulas
Subtraction: (A − B)ij = Aij − Bij
Multiplication: (AB)ij = Σ Aik · Bkj
2×2 Determinant: det(A) = ad − bc
3×3 Determinant: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Transpose: (AT)ij = Aji
Inverse: A−1 = (1 / det(A)) · adj(A)
Examples
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]
A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]]
A = [[1, 2], [3, 4]], B = [[5, 6], [7, 8]]
AB = [[1·5+2·7, 1·6+2·8], [3·5+4·7, 3·6+4·8]] = [[19, 22], [43, 50]]
A = [[3, 8], [4, 6]]
det(A) = 3·6 − 8·4 = 18 − 32 = −14
Frequently Asked Questions
What is a matrix determinant?
The determinant is a scalar value computed from a square matrix. It tells you whether the matrix is invertible (non-zero determinant = invertible) and represents the scaling factor of the linear transformation described by the matrix. For a 2×2 matrix [[a,b],[c,d]], the determinant is ad − bc.
How do you multiply two matrices?
To multiply two matrices A and B, take the dot product of each row of A with each column of B. The element at position (i,j) in the result is the sum of A[i,k] · B[k,j] for all k. The number of columns in A must equal the number of rows in B.
When is a matrix not invertible?
A matrix is not invertible (singular) when its determinant equals zero. This means the matrix maps some non-zero vector to the zero vector, and no unique inverse exists. Geometrically, the transformation collapses space into a lower dimension.