Perform operations on 2×2 and 3×3 matrices — addition, subtraction, multiplication, determinant, transpose, and inverse.
This tool performs operations on matrices — including addition, subtraction, multiplication, determinant calculation, and finding the inverse. Matrices are fundamental in linear algebra, used in computer graphics, engineering, physics, and data science.
Matrix addition/subtraction combines corresponding elements. Matrix multiplication involves multiplying rows by columns and summing the products. The determinant (for square matrices) is calculated through a formula that varies by matrix size, and the inverse (when it exists) satisfies A × A⁻¹ = the identity matrix.
Enter the dimensions and values of your matrix or matrices, select the operation you want to perform, and the calculator returns the result.
Example: Adding matrix [[1,2],[3,4]] and [[5,6],[7,8]] gives [[6,8],[10,12]], by adding each corresponding element.
Why does matrix multiplication work differently than regular multiplication? Matrix multiplication combines rows of the first matrix with columns of the second, summing products of corresponding elements — this means matrix multiplication is generally not commutative (A×B usually doesn't equal B×A), unlike regular number multiplication.
What is a determinant used for? The determinant indicates whether a matrix has an inverse (a zero determinant means no inverse exists), and it's used in solving systems of linear equations, calculating area/volume transformations, and various engineering applications.
What is the identity matrix? The identity matrix is a square matrix with 1s along the diagonal and 0s elsewhere, functioning similarly to the number 1 in regular multiplication — multiplying any matrix by the identity matrix leaves it unchanged.
Can all matrices be inverted? No, only square matrices (same number of rows and columns) with a non-zero determinant have an inverse; matrices without an inverse are called singular matrices.
Where are matrices used outside of pure math? Matrices are used extensively in computer graphics (transforming and rendering 3D objects), machine learning (representing data and neural network weights), engineering simulations, and economic modeling.