(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 $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
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 $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
- The letters of the word "UDAYPUR" are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word "UDAYPUR" is
- Let $f$ be a function such that $3f(x) + 2f\left(\frac{m}{19x}\right) = 5x, x \neq 0$, where $m = \sum_{i=1}^{9} (i)^2$. Then $f(5) - f(2)$ is equal to
- The smallest positive integral value of $a$, for which all the roots of $x^4 - ax^2 + 9 = 0$ are real and distinct, is equal to
- $\left(\frac{1}{3} + \frac{4}{7}\right) + \left(\frac{1}{3^2} + \frac{1}{3} \times \frac{4}{7} + \frac{4^2}{7^2}\right) + \left(\frac{1}{3^3} + \frac{1}{3^2} \times \frac{4}{7} + \frac{1}{3} \times \frac{4^2}{7^2} + \frac{4^3}{7^3}\right) + \dots$ upto infinite terms, is equal to
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
Important Questions from Algebra
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
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 $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
