Download matrices - GCG-42

MATRICES
A rectangular arrangement of elements is called matrix.
Types of matrices:
Null matrix:
A matrix whose all elements are zero is called a null matrix or zero matrix.
Upper triangular matrix:
In an upper triangular matrix, all elements below the main diagonal are
zero.
Lower triangular matrix:
In a lower triangular matrix, all elements above the main diagonal are
zero.
ALGEBRA OF MATRICES
PROPERTIES OF MATRIX ADDITION
(i) matrix addition is commutative
i.e. if A and B are two m x n matrices then A + B = B + A
(ii) matrix addition is associative
i.e. if A, B and C are three matrices of the same order then
(A + B) + C = A + (B +C).
(iii) Existence of identity
The null matrices is the identity element for matrix addition i.e A + O = A =
O +A.
(IV) Existence of Inverse
For every matrix A there exit a matrix –A such that A+(-A) = O.
(v) Cancellation laws hold good in case of addition of matrices
If A, B and C are three matrices of the same order then
A+B=A+C
=» B =C (left cancellation law)
B + A =C + A =» B =C (right cancellation law) MULTIPLICATION OF
MATRICES
Remark: If A and B are two matrices such that AB exists , then BA may or
may not exist.
TRANSPOSE OF A MARTIX
AT is obtained from A by changing its rows into columns and its columns
into rows.
PROPERTIES OF TRANSPOSE
Let A and B be two matrices. Then
(i) (AT)T = A
(ii) (A+B)T = AT + BT, A and B being of same order .
(iii) (kA)T = k AT, k be any scaler
(iv) (AB)T = BT AT
Symmetric Matrix:
A square matrix A is a symmetric matrix iff AT = A.
Skew-symmetric Matrix:
A square matrix A is a skew-symmetric matrix iff AT = -A.
DETERMINANT
Every square matrix can be associated to an expression or a no. known as
determinant. Denoted by |A|
Singular matrix:
A square matrix is a singular matrix if its determinant is zero. Otherwise , it
is a non-singular matrix.
PROPERTIES OF DETERMINANTS
P-1 Let A = [aij] be a square matrix of order n, then the sum of the product
of elements of any row (column) with their cofactors is always equal to |A|.
P-2 Let A = [aij ] be a square matrix of order n, then the sum of the
products of elements of any row (column) with the cofactors of the
corresponding elements of some other row (column) is zero.
P-3 Let A = [aij ] be a square matrix of order n, then
|A| = |AT|.
P-4 Let A = [aij ] be a square matrix of order n(≥2) and let B be a matrix
obtained from A by interchanging any two rows (columns) of A, then |B| = |A|.
P-5 If any two rows (columns) of a square matrix A = [aij ] be a square
matrix of order n(≥2) are identical, then its determinant is zero i.e. |A| = 0.
P-6 Let A = [aij ] be a square matrix of order n, and B be the matrix
obtained from A by multiplying each element of a row (column) of A by a
scalar k, then |B| = k |A|.
Inverse of a matrix
A square matrix of order n is invertible if there exists a square matrix B of
the same order such that
AB = In = BA.
In such a case, we say that the inverse of A is B and we write, A-1 = B.
Algorithm for finding the inverse of a square matrix A
STEP 1 Find |A|
STEP 2 If |A| = 0, then write” A is a singular matrix and hence not
invertible”
STOP
Else write “A is a non-singular & hence invertible ”.
STEP 3 Calculate the cofactors of elements of A
STEP 4 Write the matrix of cofactors of elements of A and
then obtain
its transpose to obtain adj A.
STEP 5 Find the inverse of A by using the formula
A-1 = 1 adj A
|A|
STOP