Transpose of a matrix

Chapter 3 Class 12 Matrices
Concept wise

In transpose of a matrix,

• Rows become columns and,
• Column become rows

For matrix

It’s transpose is

We denote it by A’

Similarly for

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

(A’)’ = A

Therefore,

(A’)’ = A

Therefore,

(4A)’ = 4A’

## (AB)’ = B’ A’

Therefore,

(AB)’ = B’ A’

### Transcript

For matrix A = [■8(3&2@1&4)] It’s transpose is A’ = [■8(3&1@2&4)] B = [■8(3&2@1&4@5&3)] B’ = [■8(3&1&5@2&4&3)] A = [■8(3&1@2&4)] B = [■8(−8&2@−4&0)] A = [■8(3&2@1&4)] A’ = [■8(3&1@2&4)] (A’)’ = [■8(3&1@2&4)]^′ = [■8(3&2@1&4)] = A A = [■8(3&2@1&4)] Let k = 4 (4A)’ 4A = 4 [■8(3&2@1&4)] = [■8(4×3&4×2@4×1&4×4)] = [■8(12&8@4&16)] 4A’ A = [■8(3&2@1&4)] A’ = [■8(3&1@2&4)] (4A)’ = [■8(12&8@4&16)]^′ = [■8(12&4@8&16)] 4A’ = 4[■8(3&1@2&4)] = [■8(4×3&4×1@4×2&4×4)] = [■8(12&4@8&16)] Let A = [■8(3&1@2&4)], B = [■8(−8&2@−4&0)] (A + B)’ A + B = [■8(3&2@1&4)] + [■8(−8&2@−4&0)] = [■8(3+(−8)&2+2@1+(−4)&4+0)] = [■8(−5&4@−3&4)] A’ + B’ A’ = [■8(3&1@2&4)] B’ = [■8(−8&2@−4&0)] (A + B)’ = [■8(−5&−3@4&4)] A’ + B’ = [■8(3&1@2&4)] + [■8(−8&−4@2&0)] = [■8(3+(−8)&1+(−4)@2+2&4+0)] = [■8(−5&−3@4&4)] Let A = [■8(3&1@2&4)], B = [■8(−8&2@−4&0)] (AB)’ AB = [■8(3&1@2&4)] [■8(−8&2@−4&0)] = [■8(3×(−8)+2×(−4)&3×2+2×0@1×(−8)+4×(−4)&1×2+4×8)] = [■8(−24−8&6+0@−8−16&2+0)] = [■8(−32&6@−24&2)] (AB)’ = [■8(−32&−24@6&2)] B’A’ A = [■8(3&1@2&4)] A’ = [■8(3&1@2&4)] B = [■8(−8&2@−4&0)] B’ = [■8(−8&−4@2&0)] B’A’ =[■8(−8&−4@2&0)][■8(3&1@2&4)] = [■8((−8)×3+(−4)×2&(−8)×1+(−4)×4@2×3+0×2&2×1+0×4)] = [■8(−24−8&−8−16@6+0&2+0)] = [■8(−32&−24@6&2)]

Made by

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.