Matrix Calculator — Add, Subtract, Multiply & Find Determinant
What is the matrix result?
Matrix Calculator & Analyzer
Add, subtract, multiply, transpose, find determinants and inverses for 2×2 and 3×3 matrices. Properties, steps, and charts update live as you type.
Quick Presets
Matrix Size
Operation
Matrix A
Matrix B
Results are for informational purposes only.
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Perform matrix addition, subtraction, multiplication, and scalar multiplication — plus calculate the determinant and transpose of any matrix up to 4×4 in size. Matrix operations are the foundation of linear algebra, computer graphics, machine learning, and engineering simulations. This calculator handles all core operations with clean output and shows key properties of each result: determinant, rank, whether the matrix is symmetric or singular. The determinant is especially important: a determinant of zero means the matrix is singular — it has no inverse, meaning the system of equations it represents has no unique solution. In practical terms, a singular matrix means you cannot solve the underlying linear system uniquely — critical in structural engineering and regression analysis. Matrix multiplication follows strict dimension rules: an m×n matrix times an n×p matrix yields an m×p result. This calculator validates dimensions automatically and explains any mismatch. The transpose operation flips a matrix across its diagonal — a common step in statistics and graphics transformations.
- →Completing linear algebra homework involving matrix multiplication or determinants
- →Checking whether a matrix is invertible by computing its determinant
- →Verifying hand calculations for matrix operations
- →Understanding transformation matrices in computer graphics or robotics
- →Solving systems of linear equations represented in matrix form
A student needs to multiply two 3×3 matrices for a linear algebra assignment. Matrix A = [[1,2,3],[4,5,6],[7,8,9]], Matrix B = [[9,8,7],[6,5,4],[3,2,1]]. The calculator computes each result element as a dot product of the corresponding row and column and shows the full result matrix. It also reports that the determinant of A is 0 — confirming A is singular and non-invertible.
Why does matrix multiplication not commute (A×B ≠ B×A)?
Matrix multiplication computes dot products of rows from the first matrix with columns of the second. Swapping the order changes which rows pair with which columns, producing different dot products and usually a different result. Commutativity (A×B = B×A) only holds for special cases like diagonal matrices or when one matrix is the identity. In general, always track which matrix is on the left vs. right.
What does a determinant of zero mean?
A zero determinant means the matrix is singular: its rows or columns are linearly dependent, it cannot be inverted, and the system of equations it represents either has no solution or infinitely many solutions. In geometric terms, the transformation collapses space into a lower dimension — for a 2×2 matrix, it flattens the plane into a line.
What are the dimension rules for matrix multiplication?
To multiply Matrix A (m×n) by Matrix B (p×q), the number of columns in A must equal the number of rows in B: n must equal p. The result will be an m×q matrix. For example, a 2×3 matrix times a 3×4 matrix produces a 2×4 result. If dimensions don't match, multiplication is undefined.
What is the transpose of a matrix?
The transpose swaps rows and columns: element at position (i,j) moves to position (j,i). A 3×2 matrix becomes a 2×3 matrix when transposed. Transpose is useful in many contexts: converting row vectors to column vectors, computing covariance matrices in statistics, and solving least-squares problems. A symmetric matrix equals its own transpose.
What is an identity matrix and why does it matter?
An identity matrix has 1s on the main diagonal and 0s everywhere else. Multiplying any matrix by a compatible identity matrix returns the original matrix unchanged — it's the matrix equivalent of multiplying by 1. The identity matrix is the basis for computing inverses: if A × B = Identity, then B is the inverse of A.