What are the following terms- eignvalues of matrix-entries of matrix-equality of matrix-matrix of groups-identity of matrix-inverse matrix-multiplication of matrices?

Answer 1

First, a small note: an m-by-n or m x n matrix has m rows and n columns.

• The eigenvalues λ of a matrix A are scalars such that Ax = λx for some nonzero x vector.
• The entries aij of a matrix A are the numbers contained within the matrix, each with a unique position of the ith row and jth column.
• 'Equality' in matrices has the same definition as for the rest of mathematics.
• A matrix of groups is a matrix whose entries are members of a group, often with specific entries in certain positions.
• The matrix identity In is that square n by n matrix whose entries aij are 1 if i = j, and 0 if i ≠ j.
• The inverse of a square matrix A is the square matrix B such that AB = In, denoted by B = A-1.

• Matrix multiplication is the act of combining two matrices, the p-by-q A = (aij) and the q-by-r B = (bij) to form the new matrix p-by-r C = (cij) such that cij = Σaikbkj, where 1 ≤ k ≤ q. This is denoted by C = AB. Note that matrix mulplication is not commutative, i.e. AB does not necessarily equal BA; the order of the components is important and must be maintained to achieve the result. Note also that although p does not need to equal r, q must be the same in each matrix.