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Matrices: - An M X N Matrix Is Written As

Use full for MHT-CET matrices revision, One can use memory map from a person who has produced state topper of PMT and PET same year.

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394 views7 pages

Matrices: - An M X N Matrix Is Written As

Use full for MHT-CET matrices revision, One can use memory map from a person who has produced state topper of PMT and PET same year.

Uploaded by

nitin
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PPS, PDF, TXT or read online on Scribd
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Matrices

• An m x n matrix is written as
 a11 a12 a1j a1n 
 
 a 21 a 22 a 2n 
 
 
•A=  
 ai1 ithrow → aij 
 
 
 am1 am2 amn  mxn
Properties of Matrices
• If A,B and C are matrices of same order
then
A + B = B + A Matrix addition is
commutative
(A+B) + C = A+ (B+C) Associative
A + O = A = O + A Property of null matrix
A +(-A) = O = -A + A Inverse Property
Properties of Matrices
• If A = [aij]nxm then scalar multiplication of A
and α is defined as [α .aij]nxm and denoted
as αA
• (α + β).A = α.A + β.B
∀ α.( A + B) = α.A + α.B
• (-1).A = -A
• (α • β).A = α .( β.A)
Types of matrices
• : A square matrix is
• Upper triangular if aij=0 if i > j
• Lower triangular if aij=0 if i < j
• Diagonal if is upper ∧ lower ∆ ie aij = 0 if i ≠ j
• Scalar matrix if diagonal and aij = λ if i = j
• Unit matrix if is scalar and λ = 1
• Null matrix if aij = 0
• Idempotent Matrix if A2 = A
• Periodic with period k if Ak+1 = A
• Nilpotent of order k if Ak = O
• Involutory matrix A2 = I
• Symmetric if AT = A’ = A ie aij = aji
• Skew symmetric AT = A’ = –A ie
aij = –aji
• Orthogonal if AT = A–1 ie AAT = I
• Singular |A| = 0
• Non singular |A| ≠ 0
Points to remember
• A = diag(a1 , a2, a3, - - an) then
• A–1 = diag(1/a1 , 1/a2, 1/a3, - - 1/an)
• A.B = 0 does not mean either A = 0 or
B=0
• A.B may not be equal to B.A
• If A and B are orthogonal then A.B is
orthogonal
• A + AT is symmetric
• A – AT is skew symmetric
• (AB)–1 = B–1A–1
• (AB)T = B TA T
• If B is symmetric then
• (A.B.A T) is symmetric.
• If A a c  then A–1 = 1 d − c
   
b d ad − bc  − b a 

adj(A)
• A –1
= ,adj(A)=[Aij]T,
A
• Aij = Cofactor of aij = (-1)i+j Mij
• Mij = Minor of aij = determinant by
k= p
deleting i row and j column.
th th

• [aij]mxp. [bij]pxn =[cij]mxn cij =


∑
k =1
aik bkj

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