A matrix calculator can help you work with arrays of numbers arranged in rows and columns. Matrices are used in algebra, computer graphics, statistics, and engineering. Learning the rules behind the calculations helps you recognize invalid operations and check your results.
What is a matrix?
A matrix is a rectangular arrangement of entries. A matrix with two rows and three columns has dimensions 2 × 3. The position of each entry is identified by its row and column.
How to use the matrix calculator
- Open the Matrix Calculator.
- Choose the operation supported by the tool, such as addition or multiplication.
- Enter the matrix dimensions and values in the correct positions.
- Calculate the result and check its dimensions.
- For determinants or inverses, confirm that the input matrix is square.
Matrix addition and subtraction
Two matrices can be added or subtracted only when they have the same dimensions. Add corresponding entries. For example, if A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then A + B = [[6, 8], [10, 12]].
Matrix multiplication
To multiply A by B, the number of columns in A must equal the number of rows in B. The result has the row count of A and column count of B. For example, multiplying A = [[1, 2], [3, 4]] by B = [[2, 0], [1, 2]] gives AB = [[4, 4], [10, 8]]. In general, AB does not equal BA.
Transpose of a matrix
The transpose swaps rows and columns. If A = [[1, 2, 3], [4, 5, 6]], then Aᵀ = [[1, 4], [2, 5], [3, 6]]. The dimensions change from 2 × 3 to 3 × 2.
Determinant of a 2 × 2 matrix
For A = [[a, b], [c, d]], the determinant is det(A) = ad − bc. If A = [[3, 2], [1, 4]], the determinant is 3 × 4 − 2 × 1 = 10.
When does a matrix have an inverse?
A square matrix has an inverse precisely when its determinant is nonzero. For the 2 × 2 matrix [[a, b], [c, d]], the inverse is (1/(ad−bc)) × [[d, −b], [−c, a]] when ad − bc is not zero. A matrix with zero determinant is singular and has no inverse.
Common matrix operations
| Operation | Requirement |
|---|---|
| Addition | Same dimensions |
| Multiplication AB | Columns of A = rows of B |
| Transpose | Any rectangular matrix |
| Determinant | Square matrix |
| Inverse | Square matrix with nonzero determinant |
Frequently asked questions
Can I multiply a 2 × 3 matrix by a 3 × 4 matrix?
Yes. The product has dimensions 2 × 4.
Is matrix multiplication commutative?
Generally no. AB and BA can differ, and one product may be defined when the other is not.
Can a nonsquare matrix have a determinant?
No. The standard determinant is defined for square matrices.
What is the identity matrix?
A square matrix with ones on the main diagonal and zeros elsewhere. Multiplying a compatible matrix by an identity matrix leaves it unchanged.
Try the Matrix Calculator to check supported matrix operations.
Moiz Ahmad is a digital marketer and SEO expert based in Dera Ismail Khan, Pakistan. With over 6 years of experience in SEO, WordPress development, and digital strategy, he has helped 30+ businesses rank on Google’s first page. He is the founder of DigiTechPak and Moiz Blogger. Connect: LinkedIn | Twitter | Moiz Blogger