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In transpose of a matrix,

  • Rows become columns and,
  • Column become rows

For matrix

69.jpg

It’s transpose is

Transpose of a Matrix - Part 2

We denote it by A’

 

Similarly for

Transpose of a Matrix - Part 3

 

Let’s look at some properties of transpose

 

Properties of transpose of a matrix

  • (A’)’ = A
  • (kA)’ = kA’
  • (A + B)’ = A’ + B’
  • (AB)’ = B’ A’

Let’s try to prove them one by one

 

Let

Transpose of a Matrix - Part 4

 

(A’)’ = A

Transpose of a Matrix - Part 5

Therefore,

  (A’)’ = A

 

 

Transpose of a Matrix - Part 6

 

Therefore,

  (4A)’ = 4A’

 

(A + B)’ = A’ + B’

Transpose of a Matrix - Part 7

 

(AB)’ = B’ A’

Transpose of a Matrix - Part 8

 

Therefore,

(AB)’ = B’ A’

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Transcript

For matrix A = [■8(3&[email protected]&4)] It’s transpose is A’ = [■8(3&[email protected]&4)] B = [■8(3&[email protected]&[email protected]&3)] B’ = [■8(3&1&[email protected]&4&3)] A = [■8(3&[email protected]&4)] B = [■8(−8&[email protected]−4&0)] A = [■8(3&[email protected]&4)] A’ = [■8(3&[email protected]&4)] (A’)’ = [■8(3&[email protected]&4)]^′ = [■8(3&[email protected]&4)] = A A = [■8(3&[email protected]&4)] Let k = 4 (4A)’ 4A = 4 [■8(3&[email protected]&4)] = [■8(4×3&4×[email protected]×1&4×4)] = [■8(12&[email protected]&16)] 4A’ A = [■8(3&[email protected]&4)] A’ = [■8(3&[email protected]&4)] (4A)’ = [■8(12&[email protected]&16)]^′ = [■8(12&[email protected]&16)] 4A’ = 4[■8(3&[email protected]&4)] = [■8(4×3&4×[email protected]×2&4×4)] = [■8(12&[email protected]&16)] Let A = [■8(3&[email protected]&4)], B = [■8(−8&[email protected]−4&0)] (A + B)’ A + B = [■8(3&[email protected]&4)] + [■8(−8&[email protected]−4&0)] = [■8(3+(−8)&[email protected]+(−4)&4+0)] = [■8(−5&[email protected]−3&4)] A’ + B’ A’ = [■8(3&[email protected]&4)] B’ = [■8(−8&[email protected]−4&0)] (A + B)’ = [■8(−5&−[email protected]&4)] A’ + B’ = [■8(3&[email protected]&4)] + [■8(−8&−[email protected]&0)] = [■8(3+(−8)&1+(−4)@2+2&4+0)] = [■8(−5&−[email protected]&4)] Let A = [■8(3&[email protected]&4)], B = [■8(−8&[email protected]−4&0)] (AB)’ AB = [■8(3&[email protected]&4)] [■8(−8&[email protected]−4&0)] = [■8(3×(−8)+2×(−4)&3×2+2×[email protected]×(−8)+4×(−4)&1×2+4×8)] = [■8(−24−8&[email protected]−8−16&2+0)] = [■8(−32&[email protected]−24&2)] (AB)’ = [■8(−32&−[email protected]&2)] B’A’ A = [■8(3&[email protected]&4)] A’ = [■8(3&[email protected]&4)] B = [■8(−8&[email protected]−4&0)] B’ = [■8(−8&−[email protected]&0)] B’A’ =[■8(−8&−[email protected]&0)][■8(3&[email protected]&4)] = [■8((−8)×3+(−4)×2&(−8)×1+(−4)×[email protected]×3+0×2&2×1+0×4)] = [■8(−24−8&−8−[email protected]+0&2+0)] = [■8(−32&−[email protected]&2)]

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.