Matrices
Matrix
A matrix is a rectangular arrangement of numbers (real or complex)
which may be represented as
a11 a12 a13 ... a1n
a a22 a23 ... a2n
A = 21 .
... .... .... ... ...
a ... amn
m 1 am 2 am 3
Matrix is enclosed by [ ] or ( ).
Compact form the above matrix is represented by [aij ]m × n or A = [aij ].
Element of a Matrix
The numbers a11 , a12 ,K etc., in the above matrix are known as the
element of the matrix, generally represented as aij , which denotes
element in ith row and jth column.
Order of a Matrix
In above matrix has m rows and n columns, then A is of order m × n.
Types of Matrices
(i) Row Matrix A matrix having only one row and any number of
columns is called a row matrix.
(ii) Column Matrix A matrix having only one column and any
number of rows is called column matrix.
(iii) Null/Zero Matrix A matrix of any order, having all its elements
are zero, is called a null/zero matrix, i.e. aij = 0, ∀ i , j.
(iv) Square Matrix A matrix of order m × n, such that m = n, is
called square matrix.
(v) Diagonal Matrix A square matrix A = [aij ]m × n is called a
diagonal matrix, if all the elements except those in the leading
diagonals are zero, i.e. aij = 0 for i ≠ j. It can be represented as
A = diag [a11 a22K ann ].
, (vi) Scalar Matrix A square matrix in which every non-diagonal
element is zero and all diagonal elements are equal, is called
scalar matrix, i.e. in scalar matrix, aij = 0, for i ≠ j and aij = k, for
i = j.
(vii) Unit/Identity Matrix A square matrix, in which every
non-diagonal element is zero and every diagonal element is 1, is
called unit matrix or an identity matrix,
0, when i ≠ j
i.e. aij =
1, when i = j
(viii) Rectangular Matrix A matrix of order m × n, such that m ≠ n,
is called rectangular matrix.
(ix) Horizontal Matrix A matrix in which the number of rows is less
than the number of columns, is called horizontal matrix.
(x) Vertical Matrix A matrix in which the number of rows is
greater than the number of columns, is called vertical matrix.
(xi) Upper Triangular Matrix A square matrix A = [aij ]n × n is
called a upper triangular matrix, if aij = 0, ∀ i > j.
(xii) Lower Triangular Matrix A square matrix A = [aij ]n × n is
called a lower triangular matrix, if aij = 0, ∀ i < j.
(xiii) Submatrix A matrix which is obtained from a given matrix by
deleting any number of rows or columns or both is called a
submatrix of the given matrix.
(xiv) Equal Matrices Two matrices A and B are said to be equal, if
both having same order and corresponding elements of the
matrices are equal.
(xv) Principal Diagonal of a Matrix In a square matrix, the
diagonal from the first element of the first row to the last
element of the last row is called the principal diagonal of a
matrix.
1 2 3
e.g. If A = 7 6 5 , then principal diagonal of A is 1, 6, 2.
1 1 2
(xvi) Singular Matrix A square matrix A is said to be singular
matrix, if determinant of A denoted by det (A) or| A| is zero, i.e.
|A|= 0, otherwise it is a non-singular matrix.
, Algebra of Matrices
1. Addition of Matrices
Let A and B be two matrices each of order m × n. Then, the sum of
matrices A + B is defined only if matrices A and B are of same order.
If A = [aij ]m × n and B = [bij ]m × n . Then, A + B = [aij + bij ]m × n .
Properties of Addition of Matrices
If A, B and C are three matrices of order m × n , then
(i) Commutative Law A + B = B + A
(ii) Associative Law ( A + B) + C = A + ( B + C )
(iii) Existence of Additive Identity A zero matrix (0) of order
m × n (same as of A), is additive identity, if
A+ 0= A= 0+ A
(iv) Existence of Additive Inverse If A is a square matrix, then
the matrix (– A) is called additive inverse, if
A + ( − A) = 0 = ( − A) + A
(v) Cancellation Law A + B = A + C ⇒ B = C [left cancellation law]
B + A = C + A ⇒ B = C [right cancellation law]
2. Subtraction of Matrices
Let A and B be two matrices of the same order, then subtraction of
matrices, A − B, is defined as
A − B = [aij − bij ] m × n ,
where A = [aij ]m × n , B = [bij ]m × n
3. Multiplication of a Matrix by a Scalar
Let A = [aij ]m × n be a matrix and k be any scalar. Then, the matrix
obtained by multiplying each element of A by k is called the scalar
multiple of A by k and is denoted by kA, given as
kA = [kaij ]m × n
Properties of Scalar Multiplication
If A and B are two matrices of order m × n, then
(i) k( A + B) = kA + kB
(ii) ( k1 + k2 ) A = k1 A + k2 A
(iii) k1k2 A = k1( k2 A) = k2( k1 A)
(iv) ( − k) A = − ( kA) = k( − A)
Matrix
A matrix is a rectangular arrangement of numbers (real or complex)
which may be represented as
a11 a12 a13 ... a1n
a a22 a23 ... a2n
A = 21 .
... .... .... ... ...
a ... amn
m 1 am 2 am 3
Matrix is enclosed by [ ] or ( ).
Compact form the above matrix is represented by [aij ]m × n or A = [aij ].
Element of a Matrix
The numbers a11 , a12 ,K etc., in the above matrix are known as the
element of the matrix, generally represented as aij , which denotes
element in ith row and jth column.
Order of a Matrix
In above matrix has m rows and n columns, then A is of order m × n.
Types of Matrices
(i) Row Matrix A matrix having only one row and any number of
columns is called a row matrix.
(ii) Column Matrix A matrix having only one column and any
number of rows is called column matrix.
(iii) Null/Zero Matrix A matrix of any order, having all its elements
are zero, is called a null/zero matrix, i.e. aij = 0, ∀ i , j.
(iv) Square Matrix A matrix of order m × n, such that m = n, is
called square matrix.
(v) Diagonal Matrix A square matrix A = [aij ]m × n is called a
diagonal matrix, if all the elements except those in the leading
diagonals are zero, i.e. aij = 0 for i ≠ j. It can be represented as
A = diag [a11 a22K ann ].
, (vi) Scalar Matrix A square matrix in which every non-diagonal
element is zero and all diagonal elements are equal, is called
scalar matrix, i.e. in scalar matrix, aij = 0, for i ≠ j and aij = k, for
i = j.
(vii) Unit/Identity Matrix A square matrix, in which every
non-diagonal element is zero and every diagonal element is 1, is
called unit matrix or an identity matrix,
0, when i ≠ j
i.e. aij =
1, when i = j
(viii) Rectangular Matrix A matrix of order m × n, such that m ≠ n,
is called rectangular matrix.
(ix) Horizontal Matrix A matrix in which the number of rows is less
than the number of columns, is called horizontal matrix.
(x) Vertical Matrix A matrix in which the number of rows is
greater than the number of columns, is called vertical matrix.
(xi) Upper Triangular Matrix A square matrix A = [aij ]n × n is
called a upper triangular matrix, if aij = 0, ∀ i > j.
(xii) Lower Triangular Matrix A square matrix A = [aij ]n × n is
called a lower triangular matrix, if aij = 0, ∀ i < j.
(xiii) Submatrix A matrix which is obtained from a given matrix by
deleting any number of rows or columns or both is called a
submatrix of the given matrix.
(xiv) Equal Matrices Two matrices A and B are said to be equal, if
both having same order and corresponding elements of the
matrices are equal.
(xv) Principal Diagonal of a Matrix In a square matrix, the
diagonal from the first element of the first row to the last
element of the last row is called the principal diagonal of a
matrix.
1 2 3
e.g. If A = 7 6 5 , then principal diagonal of A is 1, 6, 2.
1 1 2
(xvi) Singular Matrix A square matrix A is said to be singular
matrix, if determinant of A denoted by det (A) or| A| is zero, i.e.
|A|= 0, otherwise it is a non-singular matrix.
, Algebra of Matrices
1. Addition of Matrices
Let A and B be two matrices each of order m × n. Then, the sum of
matrices A + B is defined only if matrices A and B are of same order.
If A = [aij ]m × n and B = [bij ]m × n . Then, A + B = [aij + bij ]m × n .
Properties of Addition of Matrices
If A, B and C are three matrices of order m × n , then
(i) Commutative Law A + B = B + A
(ii) Associative Law ( A + B) + C = A + ( B + C )
(iii) Existence of Additive Identity A zero matrix (0) of order
m × n (same as of A), is additive identity, if
A+ 0= A= 0+ A
(iv) Existence of Additive Inverse If A is a square matrix, then
the matrix (– A) is called additive inverse, if
A + ( − A) = 0 = ( − A) + A
(v) Cancellation Law A + B = A + C ⇒ B = C [left cancellation law]
B + A = C + A ⇒ B = C [right cancellation law]
2. Subtraction of Matrices
Let A and B be two matrices of the same order, then subtraction of
matrices, A − B, is defined as
A − B = [aij − bij ] m × n ,
where A = [aij ]m × n , B = [bij ]m × n
3. Multiplication of a Matrix by a Scalar
Let A = [aij ]m × n be a matrix and k be any scalar. Then, the matrix
obtained by multiplying each element of A by k is called the scalar
multiple of A by k and is denoted by kA, given as
kA = [kaij ]m × n
Properties of Scalar Multiplication
If A and B are two matrices of order m × n, then
(i) k( A + B) = kA + kB
(ii) ( k1 + k2 ) A = k1 A + k2 A
(iii) k1k2 A = k1( k2 A) = k2( k1 A)
(iv) ( − k) A = − ( kA) = k( − A)
