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Question

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2×2 matrix such that the trace of A is 3 and the trace of A3 is 18, then the value of the determinant of A is

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Solution

Let A=[abcd] (1)
Given Tr(A)=3
a+d=3
d=3a
Putting in equation (1),
A=[abc3a]

Also, Tr(A)3=18
A3=[abc3a][abc3a][abc3a]
A3=[a2+bc3b3ccb+(3a)2][abc3a]
A3=[a3+abc+3bca2b+b2c+9b3ab3ac+c2b+c(3a)23bc+cb(3a)+(3a)3]
Tr(A3)=a3+abc+3bc+3bc+3bcabc+(3a)3=18
a3+9bc+(3a)3=18
a3+9bc+27a333a(3a)=18
a23a+bc=5 (2)

Now, |A|=a(3a)bc
=3aa2bc=5 [From equation (2)]

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