In the given figure, the vectors u and v are related as: Au = v by a transformation matrix A. The correct choice of matrix A is
\( \begin{bmatrix} \frac{4}{5} & \frac{3}{5} \\-\frac{3}{5} & \frac{4}{5} \end{bmatrix}\)
To find the correct transformation matrix \( A \) that relates vectors \( \mathbf{u} \) and \( \mathbf{v} \), we need to express vector \( \mathbf{v} \) as a transformation of vector \( \mathbf{u} \) using the matrix \( A \). The relationship is given as:
\( A \mathbf{u} = \mathbf{v} \).
From the diagram:
- \( \mathbf{u} = \begin{bmatrix} 4 \\ 3 \end{bmatrix} \)
- \( \mathbf{v} = \begin{bmatrix} 5 \\ 0 \end{bmatrix} \)
We need to find \( A \) such that:
\[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} 4 \\ 3 \end{bmatrix} = \begin{bmatrix} 5 \\ 0 \end{bmatrix} \]
This yields the following equations:
- \( 4a + 3b = 5 \)
- \( 4c + 3d = 0 \)
These are two linear equations in terms of the elements of matrix \( A \).
Solving these equations gives:
\[ \text{From Equation 1:} \quad a = \frac{4}{5}, \quad b = \frac{3}{5}. \] \[ \text{From Equation 2:} \quad c = -\frac{3}{5}, \quad d = \frac{4}{5}. \]
Thus, the matrix \( A \) is:
\( \begin{bmatrix} \frac{4}{5} & \frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \end{bmatrix} \)
Checking the options given, this corresponds to the first option provided in the question.
Therefore, the correct choice of matrix \( A \) is:
| \( \begin{bmatrix} \frac{4}{5} & \frac{3}{5} \\ -\frac{3}{5} & \frac{4}{5} \end{bmatrix} \) |
Important Questions from Algebra (Notes)
- What is the remainder when 2023²⁰²⁴ + 2025²⁰²⁴ is divided by 2024?
In an examination, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. If she/he attempts all 60 questions and secures 130 marks, the number of questions she/he attempts wrongly, are?
Match List-I with List-II
List-1 List-II (A) If $\begin{bmatrix}\lambda-1 & 0 \\ 0 & \lambda-1 \end{bmatrix} $, then $\lambda$ is (I) 0 (B) If A=$ \begin{bmatrix}1 & 2 \\2 & 4 \end{bmatrix} $, then $\Delta$ is (II) 1 (C) If A = $ \begin{bmatrix}1 & 0 \\0 & \frac{1}{2} \end{bmatrix} $, then $|A^{-1}|$ is (III) -2 (D) If $ \begin{bmatrix}a+1 & 1 \\1 & 2 \end{bmatrix} = \begin{bmatrix}-1 & 1 \\1 & 2 \end{bmatrix} $, then a is (IV) 2 Choose the correct answer from the options given below:
- If (x - 1) is a factor of $2x^2 - 5x + k = 0$, then the value of k is:
- If $x = (2+\sqrt{3})^{\frac{1}{3}} + (2+\sqrt{3})^{-\frac{1}{3}}$ and $x^3-3x + k = 0$, then the value of k is:
