Matrics

A matrix is a rectangular array of numbers for which operations such as addition and multiplication are defined.
Real matrices are matrices that contain real numbers.
Complex matrices are matrices that contain complex number.
Excel > Functions > Matrix Category

Rectangular array of numbers arranged in rows and columns
Matrices of the same size can be added or subtracted element by element
When using the MDETERM, MINVERSE and MMULT functions every cell in the array must contain a numeric value. If not the function will return #VALUE!
A matrix with "m" rows and "n" columns is said to be of order (m * n). When "m" and "n" are equal then the matrix is said to be square.
Two matrices are said to be identical if every element in one matrix equals the corresponding element in the other matrix.

A square matrix has an inverse if and only if its determinant is non zero
sub matrix
row vector
column vector

Identity

An identity matrix is a square matrix which contains all zeros except the main diagonal contains ones

Inverse

The inverse of a matrix is the matrix which when multiplied together gives the identity matrix.
To prove that these are in fact the correct inverses we need to multiply the two matrices together. This is shown below.
A matrix that has no inverse has a determinant of zero and is said to be singular.
The MINVERSE function returns the inverse of the matrix as an array formula.

Addition

Multiplication

Matrix multiplication can only take place when the number of columns in the first matrix equals the number of rows in the second matrix.
Matrix multiplication is not commutative so (A * B) does not equal (B * A).
Scalar multiplication
Matrix multiplication
This is usually entered as an array function and therefore must be entered with (Ctrl + Shift + Enter).
Matrix A multiplied by Matrix B is not the same as Matrix B multiplied by Matrix A (i.e. matrix multiplication is not commutative).
It is possible to use matrices to solve linear equations.
The MMULT function returns the product of two matrices.

Division

Matrix division is undefined.

Determinant

This function only has an accuracy of 16 digits so a singular array may return a result that differs by 1E-16.
The MDETERM function returns the matrix determinant as an array formula.

Transpose

This is usually entered as an array function and therefore must be entered with (Ctrl + Shift + Enter).
The first row of the input array becomes the first column of the output array.
The TRANSPOSE function will transpose a matrix and return an array formula.

Linear Equations

Matrices can be used as a concise way to work with multiple linear equations.

Linear Transformations

Symmetric

Eigenvalues and Eigenvectors

These help us to understand the geometry of linear transformations.

Decomposition

This is the process of rendering a matrix into a more easily accessible form.
Also known as Matrix Factorization

Decomposition - Cholesky

This is the decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose.