1.Algebra Booster

7.2 Algebra Booster
(vii) Identity matrix
(xv) Equality of two matrices
The sum of the diagonal elements of a matrix is known as the
Èa11 a12
˘
trace of a matrix.
Thus, if A = Í
a21 a ˙
Î 22 ˚ and Èb11 b12
˘
B = Í
b21 b ˙
Î 22 ˚,
Èa11 a12 a13
˘
If A= Í
a21 a22 a
˙
Í 23 ˙
, then Tr(A) = a 11
+ a 22
+ a
Èa11b11 + a12b21 a11b12 + a12b22
˘
33 then AB = Í
Ía31 a32 a33
˙
a21b11 a22b21 a21b12 a22b
˙
Î + + 22 ˚
Î
˚
In a scalar matrix, if all the diagonal elements are 1, it is
called an identity matrix.
Two comparable matrices are said to be equal if their corresponding
elements are the same.
1 0 0
For example, I
1 0 È ˘
È2 3˘ Èa
b˘
È ˘ , 0 1 0 ,etc.
2= I
Í ˙
If Í ,
Í 3=
0 1
˙
4 6
˙ = Í
c d
˙ then a = 2, b = 3, c = 4 and d = 6.
Í ˙
Î ˚ Î ˚
Î ˚
ÍÎ0 0 1˙˚
(xvi) Sub-matrix
(viii) Non-zero matrix
Any matrix is obtained by eliminating some rows and some
In a matrix, if at-least one element is non-zero, it is called a columns from a given matrix A, it is called a sub-matrix of A.
non-zero matrix.
È1 3 4 6˘
1 2 0 0
For example, A= Í ˘ and B =
Ê ˆ
Let A
Í
7 0 5 2
˙ 1 3 4
= Í ˙
and B = È ˘
Í
7 0 5 ˙
3 4
˙ Á
Ë0 2˜
Î ˚
¯
Í2 5 9 0˙
Î ˚,
Î ˚
are non-zero matrices.
(ix) Zero matrix
In a matrix, if every elements are zero, it is known as zero
matrix. It is denoted as O.
B is a sub-matrix of A.
3. ADDITION OF MATRICES
We can find the addition of two or more matrices if they are
comparable matrices otherwise addition is not defined.
È0 0˘ È0 0 0˘
For example, O = Í , O = , etc.
0 0
˙ Í
0 0 0
˙
1 3 2 3
Î ˚ Î ˚
Let A= Í ˘ and
6 5
˙ B =
È Í ˘
4 1
˙
(x) Upper triangular matrix
Î ˚ Î ˚,
In a square matrix, if all the elements below the leading elements
are zero, it is called a upper triangular matrix.
then A + B = Í
È 3 6˘
10 6
˙
Î ˚
1 2 3
For example, A
1 2 È ˘
È ˘
= , B =
Í 0 4 5 ˙
Í ,etc.
0 4
˙ Í ˙
Properties of Addition of Matrices
Î ˚
ÍÎ0 0 7˙
(i) Matrix addition is commutative.
˚
(ii) Matrix addition is associative.
(xi) Lower triangular matrix
(iii) Additive identity of a matrix exists.
In a square matrix, if all the elements above the leading elements
are zero, it is called a lower triangular matrix.
(iv) Additive inverse of a matrix exists.
1 0 0
For example, A
1 0 È ˘
3.1 Scalar Multiplication
È ˘
= , B =
Í 2 7 0 ˙
Í ,etc.
2 7
˙ Í ˙
2 4
Î ˚
If A = È ˘
ÍÎ2 3 4˙
Í
˚
6 7 ˙
Î ˚, then È2k
4k˘
kA = Í
6k
7k
˙
Î ˚,
(xii) Strictly triangular matrix
where k is the scalar multiple of A.
In a square matrix, if all the diagonal matrices are zero, it is
called a strictly triangular matrix.
3.2 Negative of a matrix
0 2 3
For example, A
0 3 È ˘
2 4
È ˘
= , B =
Í 5 0 7 ˙
If A = È ˘
Í ,etc.
Í
2 0
˙ Í ˙
6 7 ˙
Î ˚, then È-2 -4˘
- A = Í
-5 -7
˙
Î ˚
Î ˚
ÍÎ3 6 0˙˚
(xiii) Comparable matrices
4. MULTIPLICATION OF MATRICES
Two matrices are said to be comparable matrices, if their orders
are the same.
If the number of columns of a first matrix is equal to the number
of rows of a second matrix, we can find out the product,
È1 2˘ Èa
b˘
otherwise product is not defined.
Let A= Í and B =
3 4
˙ Í
c d
˙ are two comparable matrices.
Î ˚ Î ˚ If A= [ a ij ] m¥
n and B = [ b ji ] n¥
p ,
(xiv) Trace of a matrix
then AB = [ c ij ] m¥
p