Matrix Algebra & Linear Equations Solver

Choose a mode (Matrix Operations or Solve Linear System) and a matrix size (2×2 or 3×3), then enter the matrix entries into the grid. For system solving, also enter the constant vector b. The calculator returns the determinant, inverse, transpose, or solution vector along with the intermediate steps and a geometric visualization of how the matrix transforms the basis vectors.

Consider the system Ax = b with A = 2 1 -1 ; -3 -1 2 ; -2 1 2 and b = 8 ; -11 ; -3 , this calculator's default inputs.

det(A) = -1

Replacing each column of A with b in turn gives det(A_x) = -2, det(A_y) = -3, and det(A_z) = 1.

x = (-2)/(-1) = 2, y = (-3)/(-1) = 3, z = 1/(-1) = -1

This is a classic textbook system, confirming x=2, y=3, z=-1.

To solve the same type of linear system with two unknowns directly, use the system of equations calculator. For vector operations like dot and cross products that pair with matrix work, see the vector math calculator.

The determinant of a square matrix, det(A) or |A|, is a scalar value that measures the geometric scaling factor of the linear transformation the matrix represents. For a 2 × 2 matrix A = a b ; c d , the determinant is det(A) = ad - bc. For a 3 × 3 matrix, this calculator uses cofactor expansion along the first row: det(A) = A_(11)(A_(22)A_(33) - A_(23)A_(32)) - A_(12)(A_(21)A_(33) - A_(23)A_(31)) + A_(13)(A_(21)A_(32) - A_(22)A_(31))

A matrix is invertible (non-singular) if and only if det(A) ≠ 0. If det(A) = 0, the matrix is singular and no inverse exists. This calculator detects that case and reports it rather than dividing by zero.

The inverse matrix A^(-1) satisfies A A^(-1) = A^(-1) A = I, where I is the identity matrix. This calculator computes it via the adjugate (the transpose of the cofactor matrix C): A^(-1) = 1/det(A) adj(A) = 1/det(A) C^T

Cramer's Rule solves linear systems A x = b directly using determinants, replacing column i of matrix A with the constant vector b to find variable x_i: x_i = det(A_i)/det(A) This is efficient for the 2×2 and 3×3 systems this calculator handles, though for larger systems Gaussian elimination scales better. See "Beyond This Calculator" below.

Matrices are the computational backbone of computer graphics, machine learning, and physical simulations. In 3D graphics and game engines, geometric transformations (translation, rotation, scaling) are executed with transformation matrices: multiplying a vertex position by a matrix repositions it in space. In machine learning, neural network layers are chains of matrix multiplications combined with non-linear activation functions.

In electrical engineering, nodal analysis uses matrices to solve circuit networks; in civil engineering, stiffness matrices model stress and strain across structures. The same determinant, inverse, and system-solving building blocks this calculator demonstrates on small 2×2/3×3 systems scale up (with more efficient algorithms) to power all of these applications.

A few related linear-algebra concepts come up often alongside matrix work but aren't computed by this tool. Eigenvalues and eigenvectors, scalars λ and vectors v satisfying Av = λ v, found by solving det(A - λ I) = 0, underlie PageRank, PCA, and vibration analysis, but this calculator doesn't solve characteristic polynomials or return eigenvalues. Gaussian elimination and LU decomposition are the standard ways to solve larger systems efficiently (Cramer's rule becomes impractical past 3×3 or 4×4 due to its factorial-growth cost); this calculator uses Cramer's rule directly since it's limited to 3×3. Matrix condition numbers, which measure numerical sensitivity to small input errors, also aren't computed here.

If your work needs any of those (larger systems, eigenvalue decomposition, or numerical stability analysis), a dedicated linear algebra library (NumPy, MATLAB, R) is the right tool; this calculator is built for quick, exact answers on small 2×2/3×3 problems with the full worked steps shown.