Fundamental of Matrices
* A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.
Column matrix: A matrix is said to be column matrix if it has only one column.
Row matrix: A matrix is said to be a row matrix if it has only one row.
Square matrix: A matrix, in which the number of rows is equal to the number of columns, is said to be a square matrix.
Diagonal matrix: A square matrix is said to be a diagonal matrix if all its non diagonal elements are zero.
Scalar matrix: A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal.
Identify matrix: A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix.
Zero matrix: A matrix is said to be zero matrix or null matrix if all its elements are zero.
Addition and Subtraction of Matrices
* If X = [x_{ij} ], and Y =[ y_{ij}] are two matrices of the same order, say m × n, then the sum, X + Y, is defined as the matrix Z = [ z_{ij} ] of the other m × n,
where [ z_{ij} ] = [ x_{ij} ] + [ y_{ij} ], for all values of I and j.
* If A = [ a_{ij} ] m×n is a matrix and k is a scalar, then kA is another matrix that is obtained by multiplying every element of A by the scalar, k.
* If X = [ x_{ij} ], be an m × n matrix and O be an m × n zero matrix; then X + O = O + X = X.
in other words, O is the additive identity for matrix addition.
* Let X = [ x_{ij} ], be any matrix; then we have another matrix as –X = [ x_{ij} ] m × n;
such that
X + (- X) = (- X) + X = 0. So - X is the additive inverse of A or the negative of A.
* If A = [ a_{ij} ], and B = [ b_{ij} ] are two matrices of the same order, say m × n, then the
difference, A – B, is defined as a matrix D = [ d_{ij} ],
where d_{ij} = a_{ij} − b_{ij}, for all values of i and j. in other words,
D = A – B = A + ( - 1) B, that is, the sum of matrix A and –B.
Multiplication of Matrices
* The multiplication operation for two matrices, X and Y is possible only when the number of columns of X is equal to the number of rows of Y.
* If we have two matrices, X and Y, with order m × p and p × n, respectively, then the product of the matrices is of the order, m × n.
* The multiplication operation over two matrices need not satisfy the commutative property.
XY = 0 (null matrix) may not always imply that either X = 0 or Y = 0.
* The multiplication property over matrices satisfies the associative law and the distributive law.
* Multiplicative identity exists for any square matrix.
Symmetric and Skew symmetric Matrices
* If A = [ a_{ij} ] be an m × n matrix, then the matrix obtained by interchanging the rows and the columns of A is called the transpose of A.
* The square matrix A = [ a_{ij} ] is said to be symmetric if A’ = A, that is,
[ a_{ij} ] = [ a_{ij}] for all possible values of i and j.
* The square matrix A = [ a_{ij}] is said to be a skew symmetric matrix if A’ = - A, that is,
[ a_{ij}] = - [ a_{ij}] for all possible values of i and j.
* For any square matrix A with real number entries, A + A’ is a symmetric matrix
and A - A’ is a skew symmetric matrix.
* Any square matrix can be expressed as the sum of a symmetric and skew symmetric matrix.
Elementary Operations on a Matrix
Six operations can be performed over matrices:
- Interchange operation over rows (or columns)
- Multiplication of a row (or columns) with a non-zero number
- Addition of the elements of a row (or columns) to the corresponding elements of any other row (or columns) multiplied by a non-zero number.
Inverse of a Matrix
* If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A,
and is denoted by A^{-1}.
* Only a square matrix possesses an inverse.
* If the inverse of a matrix exists, it is unique.
* If A and B are invertible matrix of the same order, then (AB^{-1}) = B^{-1} A^{-1}.