In this post, we solve selected IIT-JEE and other higher competitive examination mathematics questions in a clear and step-by-step manner. The focus is on problem-solving techniques, shortcuts, and key observations that are useful in exams. Long theoretical discussions are avoided, and only the concepts required to solve the question efficiently are explained. This approach helps students understand how to think during the exam and arrive at the correct answer in minimum time.
Solved Question (JEE Level)
Let
\[ A= \begin{bmatrix} \frac{1}{\sqrt{2}} & -2\\ 0 & 1 \end{bmatrix}, \] \[P= \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix}, \ \theta>0 \]
If \[ B = PAP^T, \; C = P^T B^{10} P, \] and the sum of diagonal elements of \(C\) is \(\dfrac{m}{n}\), where \(\gcd(m,n)=1\), find \(m+n\).
Step 1: Use Trace Invariance
The sum of diagonal elements of a matrix is its trace. Trace remains invariant under similarity transformations.
Since \(P\) is an orthogonal matrix,
\[ P^{-1} = P^T \]
Therefore,
\[ \operatorname{tr}(C) = \operatorname{tr}(P^T B^{10} P) = \operatorname{tr}(B^{10}) \]
Step 2: Relation Between \(A\) and \(B\)
Given \[ B = PAP^T, \] so \(A\) and \(B\) are orthogonally similar matrices.
Hence,
\[ \operatorname{tr}(B^{10}) = \operatorname{tr}(A^{10}) \]
Thus,
\[ \operatorname{tr}(C) = \operatorname{tr}(A^{10}) \]
Step 3: Eigenvalues of \(A\)
Matrix \(A\) is upper triangular, so its eigenvalues are the diagonal entries:
\[ \lambda_1 = \frac{1}{\sqrt{2}}, \quad \lambda_2 = 1 \]
Step 4: Eigenvalues of \(A^{10}\)
Eigenvalues of \(A^{10}\) are:
\[ \lambda_1^{10} = \left(\frac{1}{\sqrt{2}}\right)^{10} = \frac{1}{2^5} = \frac{1}{32} \]
\[ \lambda_2^{10} = 1^{10} = 1 \]
Step 5: Trace of \(A^{10}\)
Trace equals the sum of eigenvalues:
\[ \operatorname{tr}(A^{10}) = 1 + \frac{1}{32} = \frac{33}{32} \]
Final Calculation
\[ \frac{m}{n} = \frac{33}{32} \Rightarrow m+n = 33 + 32 = \boxed{65} \]
Final Answer
\[ \boxed{65} \]
Exam Shortcut
Trace is invariant under similarity. When high powers are involved, always use eigenvalues instead of direct multiplication.
0 Comments