Select Operation
Matrix A
Matrix A (2×2)
Matrix B
Matrix B (2×2)
Result
Matrix Operation Rules
Addition / Subtraction
Same dimensions required. Add/subtract corresponding elements.
Multiplication
A(m×n) × B(n×p) = C(m×p). Not commutative: AB ≠ BA.
Transpose
Aᵀ swaps rows and columns. (AB)ᵀ = BᵀAᵀ.
Determinant
Square matrices only. det(AB) = det(A)×det(B).
Inverse
Square matrix with det ≠ 0. A×A⁻¹ = I.
Scalar Multiply
Multiply every element by the scalar k.
Special Matrices
Identity Matrix (I)
1s on diagonal, 0s elsewhere
[0 1]
Zero Matrix
All elements are 0
[0 0]
Diagonal Matrix
Non-zero only on diagonal
[0 b]
Result
Result Matrix
2×2
See below for values
Matrix Operation Examples
Quick-start with common scenarios
Practice Matrix Problems
Test your skills with practice problems
Practice with 4 problems to test your understanding.
?How Do You Calculate Matrices?
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.
What is a Matrix?
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.
Key Facts About Matrices
- Matrix dimensions: rows x columns (m x n)
- Addition/subtraction: element-wise, same dimensions required
- Multiplication: (m x n) * (n x p) = (m x p)
- Determinant: scalar value, only for square matrices
- Inverse exists only if determinant is not equal to 0
- Transpose: swap rows and columns (A^T)
- Identity matrix (I): diagonal 1s, 0s elsewhere
- Used in solving systems of linear equations: Ax = b
Quick Answer
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.
Frequently Asked Questions
What is a matrix?
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.
How do I multiply matrices?
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).
What is the determinant?
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.
What is a matrix inverse?
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.
What is matrix transpose?
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: 2026-01-15
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How this works
Matrix addition, subtraction, multiplication, transpose, determinant (via LU decomposition for n > 3), and inverse (Gauss-Jordan elimination).
Sources
- [1]NIST Digital Library of Mathematical FunctionsAcademicdlmf.nist.govAccessed Apr 21, 2026
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