Matrix Calculator

Linear algebra homework often means juggling index rules—can I multiply these sizes? is the determinant zero?—while copying numbers into a grid without a typo. This calculator handles addition, subtraction, multiplication, determinant, transpose, and inverse for matrices up to 4×4, with step-by-step output when you need to show work.

How to Use This Matrix Calculator

• Choose an operation. Pick addition, subtraction, multiplication, determinant, transpose, or inverse from the dropdown. Single-matrix ops hide Matrix B; binary ops show both grids.
• Set dimensions. Use the row and column selectors for each matrix (1 through 4). Grids rebuild when sizes change.
• Enter values. Type each cell or use Fill Random for practice matrices. Clear resets entries to zero.
• Press Calculate. The result panel shows the outcome matrix or scalar, with a step-by-step section for supported operations. For scalar arithmetic on huge integers, try the Big Number Calculator.

Matrix Formulas and Practical Applications

Matrices are tables of numbers where position matters as much as value—row and column indices tell you which entries combine during each operation.

Addition and subtraction

C_ij = A_ij ± B_ij

Dimensions must match exactly. Engineers add stiffness matrices in FEA builds; students use the same rule on 2×2 homework—add the top-left entry to the top-left entry, and so on.

Multiplication

C_ij = Σ_k A_ik × B_kj

A must have as many columns as B has rows. The product is an m×p matrix when A is m×n and B is n×p. Order matters: AB and BA are usually different—treat multiplication like function composition, not ordinary multiplication.

Determinant, transpose, inverse

Determinant applies to square matrices only; zero determinant means no inverse. Transpose flips rows and columns—common when converting between row and column vectors. Inverse A⁻¹ satisfies A × A⁻¹ = I, the identity matrix with ones on the diagonal.

Where matrices appear outside the classroom

• 2D and 3D graphics transforms (rotation, scale) stacked as multiplication chains.
• Solving small linear systems in circuit analysis or structural models.
• Feeding coefficients into the Quadratic Equation Calculator when a problem reduces to scalars.

Frequently Asked Questions

Can you multiply any two matrices?

No—the inner dimensions must align. If A is 2×3 and B is 3×4, the product is 2×4. If the middle sizes differ, multiplication is undefined.

When does a matrix have no inverse?

Square matrices with determinant zero are singular. The tool reports when inversion fails—often a sign of redundant equations in the underlying system.

What is the transpose of a matrix?

Rows become columns and columns become rows. For a 2×3 matrix, the transpose is 3×2 with entries reflected across the main diagonal swap.

What matrix sizes are supported?

Each grid supports 1 through 4 rows and columns—enough for typical intro linear algebra and small engineering problems.

Do addition and subtraction require matching dimensions?

Yes—both matrices must be the same shape. Browse all math calculators for related algebra tools.