Linear Algebra
Linear algebra works with matrices and vectors — the language of systems of equations, transformations and high-dimensional data. These tools cover the basic operations on both.
Two objects dominate linear algebra: the matrix, which packages a system of linear equations or a transformation, and the vector, which represents a point or direction. Almost every operation in the subject is a way to combine these or extract a number from them.
Determinant and invertibility
The determinant of a square matrix is a single number that tells you, among other things, whether the matrix is invertible. A zero determinant means singular — no inverse exists.
Matrix arithmetic
Matrices of the same size add and subtract entrywise. Matrices of compatible dimensions multiply by combining each row of one with each column of the other.
Matrix inverse
The inverse A⁻¹ undoes the action of A: A·A⁻¹ is the identity. The inverse calculator uses Gauss-Jordan elimination after a determinant check.
Vectors in 2D and 3D
Magnitude, dot product, cross product and angle — the vector calculator gives all four, handling the 2D and 3D cases with the right formulas.
All solvers
Frequently asked questions
What does it mean for a matrix to be singular?
A singular matrix has determinant 0; its rows are linearly dependent and it has no inverse.
When can two matrices be multiplied?
Only when the number of columns of the first matches the number of rows of the second. The product has the rows of the first and the columns of the second.
What does the dot product tell you?
It measures how aligned two vectors are: a positive value means they point similarly, zero means perpendicular, negative means opposite-ish.