Q. 274.5( 2 Votes )

# If A = <img

Answer :

Given:

To Verify: (AB)^{-1}= B^{-1}A^{-1}

Firstly, we find the (AB)^{-1}

Calculating AB

We have to find (AB)^{-1} and

Firstly, we find the adj AB and for that we have to find co-factors:

a_{11} (co – factor of 34) = (-1)^{1+1}(94) = (-1)^{2}(94) = 94

a_{12} (co – factor of 39) = (-1)^{1+2}(82) = (-1)^{3}(82) = -82

a_{21} (co – factor of 82) = (-1)^{2+1}(39) = (-1)^{3}(39) = -39

a_{22} (co – factor of 94) = (-1)^{2+2}(34) = (-1)^{4}(34) = 34

Now, adj AB = Transpose of co-factor Matrix

Calculating |AB|

= [34 × 94 – (82) × (39)]

= (3196 – 3198)

= -2

Now, we have to find B^{-1}A^{-1}

Calculating B^{-1}

Here,

We have to find A^{-1} and

Firstly, we find the adj B and for that we have to find co-factors:

a_{11} (co – factor of 6) = (-1)^{1+1}(9) = (-1)^{2}(9) = 9

a_{12} (co – factor of 7) = (-1)^{1+2}(8) = (-1)^{3}(8) = -8

a_{21} (co – factor of 8) = (-1)^{2+1}(7) = (-1)^{3}(7) = -7

a_{22} (co – factor of 9) = (-1)^{2+2}(6) = (-1)^{4}(6) = 6

Now, adj B = Transpose of co-factor Matrix

Calculating |B|

= [6 × 9 – 7 × 8]

= (54 – 56)

= -2

Calculating A^{-1}

Here,

We have to find A^{-1} and

Firstly, we find the adj A and for that we have to find co-factors:

a_{11} (co – factor of 3) = (-1)^{1+1}(5) = (-1)^{2}(5) = 5

a_{12} (co – factor of 2) = (-1)^{1+2}(7) = (-1)^{3}(7) = -7

a_{21} (co – factor of 7) = (-1)^{2+1}(2) = (-1)^{3}(2) = -2

a_{22} (co – factor of 5) = (-1)^{2+2}(3) = (-1)^{4}(3) = 3

Now, adj A = Transpose of co-factor Matrix

Calculating |A|

= [3 × 5 – 2 × 7]

= (15 – 14)

= 1

Calculating B^{-1}A^{-1}

Here,

So,

So, we get

and

∴ (AB)^{-1} = B^{-1}A^{-1}

Hence verified

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