For the matrices A and B , verify that ( AB )′ = where (i) (ii)

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Solution

(i)

The given matrices are,

A=[ 1 −4 3 ]

And,

B=[ −1 2 1 ]

Calculate AB,

AB=[ 1 −4 3 ][ −1 2 1 ] =[ −1 2 1 4 −8 −4 −3 6 3 ]

The transpose of AB is,

( AB ) ′ =[ −1 4 −3 2 −8 6 1 −4 3 ]

The transpose of A and B is,

A ′ =[ 1 −4 3 ]

And

B ′ =[ −1 2 1 ]

Calculate B ′ A ′ .

B ′ A ′ =[ −1 2 1 ][ 1 −4 3 ] =[ −1 4 −3 2 −8 6 1 −4 3 ]

Therefore, it is verified that ( AB ) ′ = B ′ A ′ .

(ii)

The given matrices are,

A=[ 0 1 2 ]

And,

B=[ 1 5 7 ]

Calculate AB,

AB=[ 0 1 2 ][ 1 5 7 ] =[ 0 0 0 1 5 7 2 10 14 ]

The transpose of AB is,

( AB ) ′ =[ 0 1 2 0 5 10 0 7 14 ]

The transpose of A and B is,

A ′ =[ 0 1 2 ]

And,

B ′ =[ 1 5 7 ]

Calculate B ′ A ′ .

B ′ A ′ =[ 1 5 7 ][ 0 1 2 ] =[ 0 1 2 0 5 10 0 7 14 ]

Therefore, it is verified that ( AB ) ′ = B ′ A ′ .

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For the matrices A and B, verify that (AB)′ = where

(i)

(ii)

For the matrices A and B, verify that (AB)' =B'A', where $A = ⎡ ⎢ ⎣ \begin{matrix} 1 - 4 3 \end{matrix} ⎤ ⎥ ⎦, B = [- 121]$

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