Matrix Calculator: Add, Subtract, Multiply, Inverse & Determinant
Free online matrix calculator: add and subtract matrices, matrix multiplication steps, scalar multiplication, and matrix inverse and determinant solver. How to add 3×3 matrices, multiply two matrices, and understand dimensions m×n. Logic Trace shows every step.
What This Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What you get:Result matrix C in a clear grid. Logic Trace: formula for each c₁₁, c₁₂, … (e.g. a₁₁ + b₁₁ for addition; dot product for multiplication). Binary: A+B, A−B, A×B, C=x·A, C=x·B, C=B. Unary: Transpose, A², determinant, inverse per matrix. Utilities: Clear, Fill (0, 1, Random), Swap A↔B, Reset. Buttons are disabled when dimensions are incompatible; inverse only when determinant ≠ 0.
- Ideal users & keywords:Students and teachers learning linear algebra, and anyone who needs a matrix determinant calculator or matrix inverse and determinant solver. High volume: matrix calculator. Long tail: matrix multiplication steps, how to add 3×3 matrices, scalar multiplication calculator, multiply two matrices, 2×2 matrix multiplication, free online matrix calculator. No sign-up; all calculations run in your browser.
- Why trust this tool:Methods follow standard linear algebra and textbook notation (m×n dimensions, row–column indexing, dot product for multiplication). Every result entry is explained in the Logic Trace—no black box. The tool is free, runs locally, and is designed for education and verification as well as quick computation.
- Scope & limits:Matrices up to 10×10. Numeric entries only. Determinant and inverse use standard algorithms; results within normal floating-point range.
This free online matrix calculator performs matrix addition, matrix subtraction, and matrix-matrix multiplication, plus unary operations: transpose, power of 2, determinant, and inverse. You get a result matrix C and a Logic Trace showing exactly which elements from A and B produced each c₁₁, c₁₂, etc. Dimensions are configurable from 1×1 up to 10×10; the tool enforces the usual constraints (same size for ±, A columns = B rows for ×).
Dimensions and Row–Column Notation (m × n)
The dimensions of a matrix are written m × n: m rows and n columns. The entry in row i, column j is denoted ai,j (and bi,j, ci,j for matrices B and C). Matrix addition and matrix subtraction require both matrices to have the same size (same m and n). Then ci,j = ai,j + bi,j or ci,j = ai,j − bi,j.
Scalar Multiplication and Matrix–Matrix Multiplication
Scalar multiplication: C = kA means every element is multiplied by k: ci,j = k · ai,j. A scalar multiplication calculator (or this tool with the scalar input) does this; dimensions do not change. Matrix-matrix multiplication AB: the number of columns of A must equal the number of rows of B. Then ci,j is the dot product of row i of A and column j of B. The matrix multiplication steps are: for each (i, j), multiply corresponding entries of row i of A and column j of B, then add. This calculator shows those steps in the Logic Trace; hovering a result cell in AB mode highlights the row of A and column of B used.
How to Add 3×3 Matrices (and Any Same-Size Matrices)
How to add 3×3 matrices: ensure both matrices are 3×3, then add element-wise: ci,j = ai,j + bi,j for each i, j. For example, the (1,1) entry is c₁₁ = a₁₁ + b₁₁; the (2,3) entry is c₂₃ = a₂₃ + b₂₃. The same rule applies to 2×2, 10×10, or any m×n pair with matching dimensions. In this matrix calculator, set both A and B to 3×3, enter values, choose A+B, and use the Logic Trace to see each sum.
Transpose, Determinant, and Inverse
Transpose: rows become columns; (AT)i,j = aj,i. Determinant: defined only for square matrices; it is a single number (det A). If det A ≠ 0, A has an inverse A−1 with A A−1 = A−1 A = I (the identity matrix). This tool acts as a matrix inverse and determinant solver: use the Determinant and Inverse buttons for each matrix. If the determinant is zero, the matrix is singular and the inverse does not exist.
Matrix Calculator FAQ
? What are the dimensions of a matrix?
Dimensions are rows × columns (m×n): m rows, n columns. Entry in row i, column j is ai,j. This matrix calculator supports 1×1 up to 10×10; the article below explains notation and when dimensions must match for each operation.
? Why do matrices have to be the same size for addition?
Addition is element-wise: ci,j = ai,j + bi,j, so every position in A must have a matching position in B. Different sizes mean no unique pairing—the calculator disables ± when dimensions differ.
? What is scalar multiplication?
Every entry is multiplied by a single number k: (kA)i,j = k·ai,j. Dimensions stay the same. This tool supports scalar multiplication (C = x·A or x·B) plus matrix-matrix multiplication, transpose, determinant, and inverse.
? How do you perform matrix multiplication manually?
For AB, A’s columns must equal B’s rows. Entry ci,j is the dot product of row i of A and column j of B. The calculator’s Logic Trace shows the exact formula for each ci,j; hover a result cell to highlight the row and column used.
? Can you multiply a 2×3 matrix by a 3×2 matrix?
Yes. A 2×3 matrix has 3 columns and a 3×2 matrix has 3 rows, so the product is defined. The result is a 2×2 matrix. In general, (m×n)(n×p) gives m×p. Use the matrix multiplication option (×) and the Logic Trace to see each dot product.
? What is a matrix inverse and when does it exist?
The inverse A−1 of a square matrix A satisfies A A−1 = A−1 A = I. It exists only when det A ≠ 0. This tool computes determinant and inverse per matrix; if det = 0, the matrix is singular and has no inverse.