(S1): $[(B-A)(B+A)]^T = \begin{bmatrix} 13 & 15 \\ 7 & 10 \end{bmatrix}$
and
(S2): $\det(\text{adj}(A+B)) = -5$,
Matrix A Determination
Given $A = \begin{bmatrix} 1 & 2 \\ 1 & \alpha \end{bmatrix}$ and the equation $A^2 - 4A + I = O$. The Cayley-Hamilton theorem states that $A^2 - (\text{tr}(A))A + (\det(A))I = O$. Comparing the given equation with the theorem:
- Trace: $\text{tr}(A) = 1 + \alpha = 4$, which implies $\alpha = 3$.
- Determinant: $\det(A) = (1)(\alpha) - (2)(1) = \alpha - 2 = 1$. Substituting $\alpha = 3$, we get $3 - 2 = 1$, confirming consistency.
Therefore, matrix $A = \begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix}$.
Matrix B Determination
Given $B = \begin{bmatrix} 3 & 3 \\ \beta & 2 \end{bmatrix}$ and the equation $B^2 - 5B - 6I = O$. Applying the Cayley-Hamilton theorem ($B^2 - (\text{tr}(B))B + (\det(B))I = O$):
- Trace: $\text{tr}(B) = 3 + 2 = 5$. This matches the coefficient of $B$.
- Determinant: $\det(B) = (3)(2) - (3)(\beta) = 6 - 3\beta$. From the equation, $\det(B)$ must be $-6$.
- Equating determinants: $6 - 3\beta = -6 \implies 3\beta = 12 \implies \beta = 4$.
Therefore, matrix $B = \begin{bmatrix} 3 & 3 \\ 4 & 2 \end{bmatrix}$.
Statement (S1) Analysis
Statement (S1) claims that $[(B-A)(B+A)]^T = \begin{bmatrix} 13 & 15 \\ 7 & 10 \end{bmatrix}$.
- Calculate $B-A$: $B-A = \begin{bmatrix} 3 & 3 \\ 4 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix}$.
- Calculate $B+A$: $B+A = \begin{bmatrix} 3 & 3 \\ 4 & 2 \end{bmatrix} + \begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ 5 & 5 \end{bmatrix}$.
- Compute the product $(B-A)(B+A)$: $(B-A)(B+A) = \begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix} \begin{bmatrix} 4 & 5 \\ 5 & 5 \end{bmatrix} = \begin{bmatrix} (8+5) & (10+5) \\ (12-5) & (15-5) \end{bmatrix} = \begin{bmatrix} 13 & 15 \\ 7 & 10 \end{bmatrix}$.
- Compute the transpose: $[(B-A)(B+A)]^T = \begin{bmatrix} 13 & 15 \\ 7 & 10 \end{bmatrix}^T = \begin{bmatrix} 13 & 7 \\ 15 & 10 \end{bmatrix}$.
The calculated transpose $\begin{bmatrix} 13 & 7 \\ 15 & 10 \end{bmatrix}$ does not match the matrix $\begin{bmatrix} 13 & 15 \\ 7 & 10 \end{bmatrix}$ in statement (S1). Hence, (S1) is incorrect.
Statement (S2) Analysis
Statement (S2) claims that $\det(\text{adj}(A+B)) = -5$.
- Calculate $A+B$: $A+B = \begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} + \begin{bmatrix} 3 & 3 \\ 4 & 2 \end{bmatrix} = \begin{bmatrix} 4 & 5 \\ 5 & 5 \end{bmatrix}$.
- Calculate the determinant of $A+B$: $\det(A+B) = (4)(5) - (5)(5) = 20 - 25 = -5$.
- Apply the property $\det(\text{adj}(M)) = (\det(M))^{n-1}$ for an $n \times n$ matrix $M$. For the 2x2 matrix $A+B$: $\det(\text{adj}(A+B)) = (\det(A+B))^{2-1} = \det(A+B)$.
- Result: $\det(\text{adj}(A+B)) = -5$.
This matches the value in statement (S2). Hence, (S2) is correct.
Final Conclusion
Based on the analysis, statement (S1) is incorrect, while statement (S2) is correct. Therefore, only (S2) is correct.
Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
