Matrix Calculator — Add, Multiply, Determinant, Inverse
Perform matrix operations on 2×2 and 3×3 matrices: addition, subtraction, multiplication, determinant, inverse, and transpose with step-by-step solutions.
Matrix A
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Matrix B
Frequently Asked Questions
What is matrix multiplication?
Matrix multiplication (A × B) requires the number of columns in A to equal the number of rows in B. For 2×2 matrices, C[i][j] = sum of A[i][k] × B[k][j] for each k. The result is another matrix of the same size.
What is the determinant of a matrix?
The determinant is a scalar value computed from a square matrix. For a 2×2 matrix [[a,b],[c,d]], det = ad - bc. For a 3×3, it expands along the first row using cofactors.
What is the inverse of a matrix?
The inverse A⁻¹ of a matrix A satisfies A × A⁻¹ = I (identity matrix). A matrix is invertible only if its determinant ≠ 0. For 2×2: A⁻¹ = (1/det) × [[d,-b],[-c,a]].
What is the transpose of a matrix?
The transpose A^T swaps rows and columns. If A[i][j] = x, then A^T[j][i] = x. Transposing a 2×3 matrix gives a 3×2 matrix.
When can you add or subtract two matrices?
Two matrices can only be added or subtracted if they have the same dimensions. The operation is performed element-by-element: (A ± B)[i][j] = A[i][j] ± B[i][j].