Let A and B be two square matrices of order 2. If \( det\,(A) = 2 \), \( det\,(B) = 3 \) and \( \det \left( {(\det \,5(det\,A)B){A^2}} \right) = {2^a}{3^b}{5^c} \) for some a, b, c, \( \in \) N, then a + b + c is equal to :
Let A and B be two square matrices of order 2. If \( det\,(A) = 2 \), \( det\,(B) = 3 \) and \( \det \left( {(\det \,5(det\,A)B){A^2}} \right) = {2^a}{3^b}{5^c} \) for some a, b, c, \( \in \) N, then a + b + c is equal to :
Show Hint
- 10
- 12
- 13
- 14
The Correct Option is B
Solution and Explanation
Step 2: The outer matrix is \((300)\,A^2\) (scalar times a \(2\times2\) matrix). \(\det\left(300A^2\right)=300^2\det(A^2)=300^2(\det A)^2=90000\times4=360000\).
Step 3: \(360000=36\times10^4=2^2\cdot3^2\cdot2^4\cdot5^4=2^6\cdot3^2\cdot5^4\). So \(a=6,\ b=2,\ c=4\).
Step 4: \(a+b+c=12\).
Answer: (B) 12
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