Perform matrix operations including addition, subtraction, multiplication, transpose, determinant, and inverse calculations.
Result Matrix
2×2
See below for values
Same dimensions required. Add/subtract corresponding elements.
A(m×n) × B(n×p) = C(m×p). Not commutative: AB ≠ BA.
Aᵀ swaps rows and columns. (AB)ᵀ = BᵀAᵀ.
Square matrices only. det(AB) = det(A)×det(B).
Square matrix with det ≠ 0. A×A⁻¹ = I.
Multiply every element by the scalar k.
1s on diagonal, 0s elsewhere
All elements are 0
Non-zero only on diagonal
Result Matrix
2×2
See below for values
Quick-start with common scenarios
Test your skills with practice problems
Practice with 4 problems to test your understanding.
A matrix is a rectangular array of numbers. Operations: Addition/subtraction (element-wise, same size). Multiplication (rows by columns, inner dimensions must match). Determinant (scalar value, square matrices). Inverse (A^-1 where A*A^-1 = I). Transpose (swap rows and columns). Used in linear algebra, graphics, and systems of equations.
A matrix is a rectangular array of numbers arranged in rows and columns. Matrix algebra provides operations for manipulating these arrays, fundamental to linear algebra, computer graphics, physics, engineering, and data science. Matrix operations solve systems of equations, perform transformations, and model linear relationships.
A matrix is a rectangular array of numbers. Operations: Addition/subtraction (element-wise, same size). Multiplication (rows by columns, inner dimensions must match). Determinant (scalar value, square matrices). Inverse (A^-1 where A*A^-1 = I). Transpose (swap rows and columns). Used in linear algebra, graphics, and systems of equations.
A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used in linear algebra, computer graphics, statistics, and machine learning. A matrix with m rows and n columns is called an m×n matrix.
To multiply A×B, the number of columns in A must equal rows in B. Each element (i,j) in the result equals the dot product of row i from A and column j from B. The result has dimensions (rows of A) × (columns of B).
The determinant is a scalar value computed from a square matrix. It indicates if the matrix is invertible (det ≠ 0) and represents the scaling factor for area/volume when the matrix is used as a transformation. For 2×2: det = ad - bc.
The inverse of matrix A (written A⁻¹) is the matrix where A × A⁻¹ = I (identity matrix). Only square matrices with non-zero determinant have inverses. For 2×2: swap diagonal, negate off-diagonal, divide by determinant.
The transpose of a matrix switches rows and columns. Element at (i,j) moves to (j,i). An m×n matrix becomes n×m. The transpose of Aᵀᵀ equals A. For symmetric matrices, A = Aᵀ.
Last updated: 2025-01-15
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Result Matrix
2×2
See below for values