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�Matrix Algebra�

Definitions
A matrix is a rectangular array of numbers, parameters or variables arranged in m-rows and n-columns. Upper case letters are used to reference them. E.g A, B, C
An ELEMENT is a number within the matrix. Lower case letters are used to represent them. E.g a_ij
The ORDER or DIMENSION of a matrix is given by (m x n). A matrix of order (1 x n) is a row vector, whilst a matrix of order (m x 1) is termed as a column vector.

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A SQUARE matrix arises when m = n
A NULL matrix arises when every element in a matrix is zero.
The IDENTITY or UNIT matrix is a diagonal matrix with the leading diagonal composed of 1’s. This is denoted by I_n, where n is the order of the matrix.
The TRANSPOSE of a matrix A dented by A¹ or A^T, is formed by writing each row of matrix A as a column in the transposed matrix A¹. Each element a_ij is written as a_ji
SYMMETRIC matrix is the one that is unaltered by transposition. Hence if A = A^T , then A is symmetric. Note that the identity matrix is symmetric.

An IDEMPOTENT matrix is a symmetric matrix for which AA = A holds. Note that the identity matrix is idempotent.
Two matrices A and B are said to be CONFORMABLE, if the number of columns in A is equal the number of rows in B.
Two matrices A and B are EQUAL if they are of the same order and all corresponding elements are equal, i.e. a_ij = b_ij for all i,j.

Types of matrices
Row matrix – (2 1 3), Column matrix, Rectangular matrix – formed by different number of rows and columns (mxn); Square matrix – (m = n); Zero matrix – all elements are zeros; Upper triangular matrix – if all elements below the diagonal are zeros; Lower triangular matrix; Diagonal matrix – all elements above and below the diagonal are zeros; Scalar matrix – same as diagonal matrix but the elements in the diagonal must be same; Identity matrix; Singular matrix – a square matrix that has no inverse; Transpose matrix; Orthogonal matrix – if it verifies that A.A^T = I