Suppose A and B are two non singular matrices such that B \neq I, | Filo
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Class 12

Math

Algebra

Determinants

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Suppose and are two non singular matrices such that and . Find the least value of for .

  1. 63
Correct Answer: Option(a)
Solution:
Step 1: Given:

Step 2: To find:
Value of for which

Step 3: Formula used:

Step 4: Solution:
Consider the given expression
   ... (1)

Step 5: Aim to make RHS as independent of
Pre-multiply (1) from both sides by , we get
Post-multiply above equation from both sides by , we get
Since, it is given that  
Expression (3) reduces to
Pre-multiply this on both sides by , we get

           .....(2) 

Step 8:
Now substituting the value of from (2) in LHS of (2) itself, we get
Since, we know
   ... (3)

Step 9:
Now substituting the value of from (2) in LHS of (3), we get

Since, we know
   ...(4)

Step 10:
Looking at the pattern of (2), (3) and (4), we can say that 
is increasing by
is forming a GP as , and
is decreasing by
In the same manner, we find that when , we get and
So, the expression becomes
Since, we know
   ...(5)

Step 11:
Now, we have
Post-Multiplying this by both sides, we get
   ... (6)
Since, , (6) reduces to
.... (7)

Step 12:
Comparing (7) with the given condition where , we find that

Step 13: Result:
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