Linear algebra guides

Short, practical explanations of the operations the calculator performs — what they mean, how to do them by hand, and which method to pick.

• How to calculate the determinant of a matrix

  Four ways to compute a determinant — the 2×2 formula, the rule of Sarrus, Laplace cofactor expansion and Gaussian elimination — and when to use each one.

  6 min read

• How to find the inverse of a matrix

  Gauss-Jordan elimination on [A | I], the adjugate formula and the 2×2 shortcut — with the conditions a matrix must satisfy to be invertible.

  6 min read

• Gaussian and Gauss-Jordan elimination explained

  Row echelon form, reduced row echelon form, pivots and the three elementary row operations, worked through on a concrete matrix.

  7 min read

• What is the rank of a matrix?

  Rank as the number of independent rows, how to compute it by elimination, and what full rank means for invertibility and linear systems.

  4 min read

• How matrix multiplication works

  The row-by-column rule, why dimensions have to match, why AB ≠ BA, and the properties that do still hold.

  5 min read

• LU decomposition, step by step

  Factoring a matrix into lower and upper triangular parts, why it is faster than repeated elimination, and how Cholesky specialises it.

  5 min read