Linear algebra is the mathematics of vectors and matrices. These guides explain the operations, what they mean geometrically, and how to compute them by hand without slips.
Matrices look like tables of numbers, but each one describes a transformation: a rotation, a stretch, a projection. Keeping that picture in mind explains why matrix multiplication is not commutative, why a zero determinant means no inverse, and what an eigenvector really is.
Every guide here pairs the computation with its meaning. You will see the rules as formulas, worked 2×2 and 3×3 examples with each product written out, and checks such as multiplying a matrix by its inverse to recover the identity. The matrix, determinant, inverse, eigenvalue, dot product and cross product calculators reproduce the same steps.
Hand computation still matters even though software multiplies matrices instantly: exams test it, and understanding a 2×2 or 3×3 case by hand is what lets you interpret a computer's output for a 100×100 one. Every example here is small enough to finish on paper.
Start with matrix basics, then determinants and inverses; vectors and eigenvalues build on both.
Linear Algebra
A matrix is a rectangular grid of numbers arranged in rows and columns. Matrices store data compactly and, more importantly, represent linear transformations and systems of equations. This guide covers the notation and the core operations, with special attention to matrix multiplication.
Linear Algebra
The determinant is a single number calculated from a square matrix. It tells you whether the matrix is invertible, how the matrix scales areas and volumes, and whether it flips orientation. This guide shows every standard method for computing it, with worked examples.
Linear Algebra
The inverse of a square matrix undoes what the matrix does, just as dividing undoes multiplying. A matrix has an inverse exactly when its determinant is nonzero. This guide covers the quick 2 by 2 formula, the general Gauss-Jordan method and how inverses solve systems of equations.
Linear Algebra
An eigenvector of a matrix is a direction that the matrix only stretches or flips, never turns, and the eigenvalue is the stretch factor. They reveal the essential behaviour of a matrix and underpin diagonalization, stability analysis, Markov chains and data compression.
Linear Algebra
The dot product combines two vectors into a single number that measures how much they point in the same direction. It is the tool for finding angles between vectors, checking perpendicularity, projecting one vector onto another and computing work in physics.
Linear Algebra
The cross product multiplies two vectors in three dimensions and returns a new vector perpendicular to both. Its length equals the area of the parallelogram the two vectors span, which makes it the standard tool for normal vectors, areas, torque and volumes.