Question

# If A=[5a−b32]  and A adj A=AAT, then 5a + b is equal to

A
-1
B
5
C
4
D
13

Solution

## The correct option is B 5Given, A=[5a−b32] 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] =[5a−b32]I2=(10a+3b)I2=(10a+3b)[1001] [10a+3b0010+3b]           . . . (i) and AAT=[5a−b32][5a3−b2] AAT=[25a2+b215a−2b15a−2b13]     . . . (ii) ∵   A(adjA)=AAT ∴ [10a+3b0010a+3b]=[25a2+b215a−2b15a−2b13]         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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