Matrices and Determinants
      Matrix definition
         Square matrices
         The identity matrix
         Vectors
      The transpose of a matrix
      Basic matrix operations
         Matrix addition and subtraction
         Scalar multiplication of a matrix
         Matrix multiplication
         The matrix-vector product
Matrices and Determinants
Matrix definition
A matrix is an ordered rectangular array of elements or entries (components) arranged in rows and columns denoted by bold uppercase letters and indicated by brackets.
A matrix with m rows and n columns is called an m-by-n matrix that has an order (or dimension) of m n
The element denoted aij lies at the intersection of the ith row and the jth column, as is shown above. 
Square matrices
A matrix with same number of rows and columns (m = n) is a square matrix of order n.
The main diagonal of a square matrix is the diagonal that runs from the top left corner to the bottom right corner.
A square matrix with all components zero except those on the main diagonal is called the diagonal matrix.
The identity matrix
An n n diagonal matrix with all non-zero elements equal to 1 is the identity matrix of order n.
An upper triangular matrix is a square matrix in which all elements underneath the main diagonal are zeros.
A lower triangular matrix is a square matrix in which all elements above the main diagonal are zeros.
Square matrices for which aij = aji are called symmetric about the main diagonal.
Examples of square matrices
  The diagonal matrix of order 4,   The identity matrix of order 3,  
     
  The upper triangular matrix of order 4,   The lower triangular matrix of order 4,  
     
             The symmetric matrix of order 5,
The elements of the main diagonal are in bold.
A matrix whose all elements are zero is called the null matrix, written 0.
Vectors
A matrix with only one row or one column is called a vector.
A matrix with one row and n columns is called a row vector.
A matrix with one column and m rows is called a column vector.
Transposition
The transpose is the matrix derived from a given matrix by interchanging the rows and columns. Thus, the transpose of a matrix A of order m n is another matrix denoted AT of order n m.
Therefore, to transpose means to interchange the rows and columns of a matrix, that is,
AT = [aji], where j is the column and i is the row of matrix A = [aij].
Example:  Given a matrix A of order 3 4, find the transpose AT.
Basic matrix operations
Matrix addition and subtraction
Matrices can be added or subtracted only if they have the same dimensions.
Addition
The elements of the sum of the two matrices with the same dimensions, A and B, are equal to the sums of the corresponding elements aij + bij that is,
A + B = [aij + bij].
Subtraction
The elements of the difference of the two matrices with the same dimensions, A and B, are equal to the differences of the corresponding elements aij - bij that is,
A - B = [aij - bij].
Example:  Given are matrices, A and B, find the sum A + B and the difference A - B.
  and  
  then,  
  and    
Scalar multiplication of a matrix
Multiplication of a matrix A by a scalar c is defined as
c A = [c aij],
that is, each element of the matrix is multiplied by c. Therefore, if c = 0, the result is the null matrix.
Example:  Given the matrix   find the product  - 2 A.  
Matrix multiplication
Multiplication of two matrices is defined only if the number of columns of the first matrix equals the number of rows of the second. Matrices that satisfy this condition are said to be conformable.
If A = [aij] is a matrix of order m n and B = [bjk] is a matrix of order n p, then the product
C = AB
where C = [cik] is the m p matrix of which the ikth entries are defined by the formula
Therefore, the ikth entry of the product equals the inner product of the components of the ith row of the first matrix with the components of the kth column of the second matrix.
Example:  Given are conformable matrices, A and B, find the product AB.
  and    
then,  
or  
The matrix-vector product
If A = [aij] is an m n matrix, x = {xj} a column vector of order n, and y = {yi} a column vector of order m
then,             y = Ax     is the matrix-vector product,
where  
The column dimension of the matrix A must equal the row dimension of the vector x to obey the product definition.
Example:  Given is the matrix A of order 2 3 and the column vector x of order 3, find the product Ax.
  and    
then,    
If A is a square matrix of order n then, 
AI = IA = A 
where I is the identity matrix of order n.
Matrices, determinants and systems of linear equations contents
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