Matrix Solver
Calculate determinants, matrix inverses, products, row reduced echelon form (RREF), and eigenvalues with full intermediate steps.
Master Matrix Operations
Matrices are the foundation of linear algebra, 3D graphics, machine learning, and quantum mechanics. Our solver shows the exact row operations, cross-products, and cofactor expansions so you can learn how each formula works.
Frequently Asked Questions
How do you find the determinant of a matrix?
For a 2x2 matrix, the determinant is ad - bc. For a 3x3, you use cofactor expansion along a row or column. Our solver shows the exact steps for both.
What is an inverse matrix?
An inverse matrix is a matrix that, when multiplied by the original matrix, yields the identity matrix. A matrix must be square and have a non-zero determinant to be invertible.
How does matrix multiplication work?
To multiply matrices, you multiply the elements of each row of the first matrix by the corresponding elements of each column of the second matrix, then add the products.
What is Row Reduced Echelon Form (RREF)?
RREF is a simplified matrix form achieved through Gaussian elimination, where leading entries are 1 and all other entries in the leading column are 0.
Can it solve eigenvalues?
Yes, our matrix calculator computes the characteristic polynomial det(A - λI) = 0 and finds real and complex eigenvalues.
Why is matrix multiplication not commutative (AB ≠ BA)?
Matrix multiplication represents the composition of linear transformations, where order fundamentally changes the geometric mapping. Furthermore, for non-square matrices of dimensions m × k and k × n, AB produces an m × n matrix, while BA may not even be dimensionally defined unless m = n.
Can a non-square matrix have an inverse?
No, a true two-sided inverse only exists for square matrices (n × n) with a non-zero determinant. For non-square rectangular matrices, generalized inverses like the Moore-Penrose pseudoinverse are used to solve least-squares optimization problems.