Hg
Name: Class 12 STICK TO YOUR WALL IN STUDY AREA
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The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
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MATRICES
• A matrix is an ordered rectangular array of numbers or expressions.
• A matrix having m rows and n columns is called a matrix of order m × n.
• aij denotes the element in i
th
row and j
th
column.
Types of Matrices:
• [aij]m × 1 is a column matrix and [aij]1 × n is a row matrix.
• An m × n matrix is a square matrix if m = n.
• A = [aij]m × m is a diagonal matrix if aij = 0, when i ≠ j.
• A = [aij]n × n is a scalar matrix if aij = 0, when i ≠ j, aij = k, (k is some constant), when i = j.
• A = [aij]n × n is an identity matrix, if aij = 1, when i = j, aij = 0, when i ≠ j.
• A zero or null matrix has all its elements as zero.
• Equal Matrices: A = [aij] = [bij] = B if (i) A and B are of same order, (ii) aij = bij for all possible values of i and j.
Operations on Matrices:
• Multiplication by a scalar: kA = k[aij]m × n = [k(aij)]m × n
• Negative of a matrix: –A = (–1)A
• Subtraction: A – B = A + (–1)B
• Commutativity for Addition: A + B = B + A
• Associative Property: (A + B) + C = A + (B + C), where A, B and C are of same order.
• k(A + B) = kA + kB, where A and B are of same order, k is constant.
• (k + l) A = kA + lA, where k and l are constant.
Matrix Multiplication:
• If A = [aij]m × n and B = [bjk]n × p , then AB = C = [cik]m × p, where c
ik
=
∑
𝑎
𝑖𝑗
𝑏
𝑗𝑘
𝑛
𝑗=1
• A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC
• Matrix multiplication is not commutative. AB may or may not be equal to BA.
• Multiplication of diagonal matrices of same order is always commutative.
• If AB = 0, doesn’t necessarily mean that either A or B = 0.
Matrix Transpose:
• If A = [aij]m × n, then A’ or A
T
= [aji]n × m
• (A’)’ = A, (ii) (kA)’ = kA’, (iii) (A + B)’ = A’ + B’, (iv) (AB)’ = B’A’
Symmetric and Skew Symmetric Matrices:
• A square matrix A is a symmetric matrix if A’ = A, and is a skew symmetric matrix if A’ = –A.
• For any square matrix A, A + A’ is symmetric, and A – A’ is skew symmetric.
• Any square matrix A = sum of a symmetric and a skew symmetric matrix. A = ½ (A + A’) + ½(A – A’)
• For symmetric matrices A, B of same order, AB is a symmetric matrix if and only if AB = BA.
Elementary operations of a matrix are as follows:
• R
i
↔ R
j
or C
i
↔ C
j
• R
i
→ kR
i
or C
i
→ kC
i
• Ri → Ri + kRj or Ci → Ci + kCj
Inverse of a Matrix:
• If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by A
-1
and A is the inverse of B.
• Inverse of a square matrix, if it exists, is unique.
• (AB)
-1
= B
-1
A
-1
