Matrices and Determinants

A matrix is a rectangular array of numbers arranged in rows and columns, and it is one of the central objects of linear algebra. Matrices can be added when they share the same shape and multiplied when the inner dimensions match, with matrix multiplication corresponding to the composition of the operations they represent. Much of their power comes from the way a matrix encodes a linear transformation, mapping vectors to other vectors while preserving the operations of addition and scaling.

The determinant is a single number computed from a square matrix that summarises important properties of the transformation it represents. Geometrically it measures how the transformation scales areas or volumes, and a determinant of zero signals that the mapping collapses space into a lower dimension. A non-zero determinant guarantees that the matrix is invertible, meaning there is an inverse matrix that reverses its action; the inverse can be expressed using the matrix of cofactors, where each cofactor is built from a smaller determinant called a minor.

Eigenvalues and eigenvectors describe special directions left unchanged in orientation by a transformation, being merely stretched or compressed by the eigenvalue factor. They are essential for understanding how repeated applications of a transformation behave and appear throughout physics, engineering and data analysis. Matrices also provide a compact way to write and solve systems of linear equations, with operations such as Gaussian elimination reducing a system to a form from which the solution can be read off directly.

Frequently asked questions

What does a determinant tell you?

It measures how a transformation scales areas or volumes. A determinant of zero means the matrix is not invertible because it collapses space.

What is an inverse matrix?

It is a matrix that reverses the action of another, so multiplying a matrix by its inverse yields the identity. Only square matrices with a non-zero determinant have one.

What are eigenvalues and eigenvectors?

An eigenvector is a direction left unchanged in orientation by a transformation, and its eigenvalue is the factor by which it is stretched or shrunk along that direction.