1. Matrix Algebra
1. MATRIX
A matrix is a rectangular array of numbers. The numbers may be real or complex. It may be
represented as
The numbers a11, a12, . . ., a1n are called the elements of the matrix. In the matrix, the horizontal lines are called rows or row vectors and the vertical lines are called columns or column vectors.
The number aij indicates the element present in the ith row and jth column.
2. TYPES OF MATRICES
A matrix A = [aij ]m×n is said to be a(i) Rectangular matrix if m 6= n
(ii) Square matrix if m = n
(iii) Row matrix if m = 1
(iv) Column matrix if n = 1
(v) Null or zero matrix if aij = 0, ∀ i and j
(vi) Diagonal matrix if m = n and aij = 0, ∀ i 6= j
(vii) Scalar matrix if m = n and aij = 0, ∀ i 6= j and aii = λ(scalar) ∀ i
(viii) Unit or Identity matrix if m = n and aij = 0, ∀ i 6= j and aii = 1 ∀ i
(ix) Upper triangular matrix if m = n and aij = 0, ∀ i > j
(x) Lower triangular matrix if m = n and aij = 0, ∀ i < j
(xi) A matrix is said to be triangular if it is either lower or upper triangular matrix.
(xii) Sparse matrix if most of the elements of the matrix are zero.
(xiii) Complex matrix if atleast one element is imaginary.
3. ALGEBRA OF MATRICES
(i) Equality of Matrices:
Two matrices are said to be equal provided they are of the same order and corresponding elements are equal.(ii) Addition of Matrices:
Two matrices A and B can be added if and only if they are
of the same order and the matrix (A + B) is obtained by adding the corresponding
elements of A and B. Addition is not defined for matrices of different sizes. The
additive inverse of A, denoted by −A.
If A and B are two matrices of the same order, then the differences between A and B
is defined by A − B = A + (−B).
Properties of Addition:
If A, B and C are three matrices of the same size, then
A + B = B + A (commutative law)
(A + B) + C = A + (B + C) (Associative law)
A = A + O = O + A (Additive property of zero)
A + (−A) = O (Additive inverse)
A + B = A + C ⇒ B = C (Left cancellation law)
B + A = C + A ⇒ B = C (Right cancellation law)
(iii) Scalar Multiplication:
If A is a matrix and K is a scalar, then KA is defined as the
matrix obtained by multiplying every element of A by K.
Properties of Scalar Multiplication:
If A,B are two matrices of the same order and
k, k1, k2 are scalars, then
(k1 + k2)A = k1A + k2A
(k1k2)A = k1(k2A)
k(A ± B) = kA ± kB
(−kA) = −(kA) = k(−A)
(iv) Multiplication of Matrices:
The product of two matrices A and B is possible only
if the number of columns of A is equal to the number of rows of B and these types of
matrices are called conformable for multiplication.
Properties of Matrix Multiplication:
If A = [aij ]m×n, B = [bij]n×p and C = [cij ]p×q then
(i) In general AB 6= BA (commutative law)
(ii) (AB)C = A(BC) (Associative law)
(iii) A(B + C) = AB + AC and (B + C)A = BA + BC (Distributive law)
(iv) AB = AC ⇒ B = C (Cancellation law). It is possible only when A is non-singular
matrix.
(v) AIn = ImA = A
(vi) k(AB) = (kA)B = A(KB), where k is a scalar
(vii) If A is a square matrix, then
Am.An = Am+n ∀ m, n ∈ N
(Am)
n = Amn ∀ m, n ∈ N
4. TRACE OF A MATRIX
Let A = [aij ]n×n be a square matrix of order n.
Then the sum of the elements lying along the principal diagonal is called the trace of A and
denoted by tr(A).
Properties of Trace of Matrix:
Let A and B be any two square matrices of order n and k is a scalar. Then
(i) tr(kA) = k tr(A)
(ii) tr(A + B) = tr(A) + tr(B)
(iii) tr(A − B) = tr(A) − tr(B)
(iv) tr(AB) = tr(BA)
5. INVOLUTORY MATRIX
If a square matrix ′A′
is such that A2 = I, then A is called Involutory. For example,
NOTE:
1. Identity matrix is always Involutory.
2. A is Involutory matrix iff (A − I)(A + I) = O
6. NILPOTENT MATRIX
For any square matrix ′A′
, if there exists a positive integer m such that A^m = O, then A is a
nilpotent matrix. The index m of the nilpotent matrix A is the least positive integer such that
A^m = O.
For example, the matrix
7. TRANSPOSE OF A MATRIX
The matrix obtained by interchanging the rows and columns of a matrix A is called transpose
of A denoted by AT or A′
.
Properties of Transpose of a Matrix:
8. DETERMINANT OF A SQUARE MATRIX
Let A = [aij ]n×n be a square matrix. Then the determinant of A is denoted by det A or |A|
and defined as
The determinant has always a real finite value. If we define a 3 × 3 determinant, then it has three rows and three columns and its value is given as follows.
This is called expanding the determinant by first row. A determinant can be expanded in terms
of any row or column.
NOTE:
1. If A is a square matrix of order n, then |A| = |AT
| .
2. If A and B are two square matrices of the same order, then |AB| = |A||B|
3. If A is a square matrix of order n, then |kA| = k
n|A|, for any scalar k.
4. |An
| = (|A|)
n
Minors and Cofactors
The minor of an element in a determinant is the determinant obtained by deleting the row and
column containing that element.
The cofactor of any element in a determinant is its minor with the proper sign. The sign
of an element in the ith row and jth column is (−1)i+j
.
The cofactor of an element is usally
denoted by the corresponding capital letter.
Thus a determinant is the sum of the products of the elements of any row (or column) by
the corresponding cofactors. This is known as Laplace’s expansion.
Properties of Determinants
(i) A determinant remains unaltered if its rows and columns are interchanged.
(ii) If any two rows (or columns) of a determinant are interchanged, the determinant
changes its sign.
(iii) A determinant vanishes if two of its rows (or columns) are identical or proportional.
(iv) If each element of a row (or column) is multiplied by a scalar, then the determinant is
multiplied by that scalar.
(v) If to each element of a row (or column) be added equi-multiples of the corresponding
elements of two or more rows (or columns), the determinant remains unaltered.
9. SINGULAR AND NON SINGULAR MATRICES
A square matrix is said to be singular matrix if determinant of the matrix is zero. Otherwise,
it is called non-singular matrix.
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