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Question

If A=[5ab32] and A adj A=AAT, then 5a + b is equal to


A
-1
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B
5
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C
4
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D
13
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Solution

The correct option is B 5
Given, A=[5ab32] and A adj A=AAT
Clearly, A (adj A) = |A| I2
[ if A is square matrix of order n, then A(adj A) = (adj A) . A = |A| In]
=[5ab32]I2=(10a+3b)I2=(10a+3b)[1001]
[10a+3b0010+3b] . . . (i)
and AAT=[5ab32][5a3b2]
AAT=[25a2+b215a2b15a2b13] . . . (ii)
A(adjA)=AAT
[10a+3b0010a+3b]=[25a2+b215a2b15a2b13] using Eqs. (i) and (ii)]
15a - 2b = 0
a=2b15 . . . (iii)
and 10a + 3b = 13 . . . (iv)
On substituting the value of 'a' from Eq. (iii) in Eq. (iv),. we get
10.(2b15)+3b=13 20b+45b15=13 65b15=13 b=3
Now, substituting the value of b in Eq. (iii), we get 5a = 2
Hence, 5a + b = 2 + 3 = 5

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