The eigen values of the matrix \(\begin{bmatrix}4 & 3 \\3 & 4 \end{bmatrix} \) are
- 1
- 2
- 7
- 4
The Correct Option is A, C
Solution and Explanation
Step 1: Define the characteristic equation. The characteristic equation is derived from the determinant of \( A - \lambda I \), leading to \( \lambda^2 - 8\lambda + 7 = 0 \).
Step 2: Calculate the determinant and simplify. The determinant simplifies to \( \lambda^2 - 8\lambda + 7 \), which factors to find the eigenvalues.
Step 3: Solve for \( \lambda \). Using the quadratic formula, we find the solutions to be \( \lambda = 7 \) and \( \lambda = 1 \), confirming the eigenvalues of the matrix.
Top Questions on Matrix
Consider the balanced transportation problem with three sources \( S_1, S_2, S_3 \), and four destinations \( D_1, D_2, D_3, D_4 \), for minimizing the total transportation cost whose cost matrix is as follows:
where \( \alpha, \lambda>0 \). If the associated cost to the starting basic feasible solution obtained by using the North-West corner rule is 290, then which of the following is/are correct?
- Three students run on a racing track such that their speeds add up to 6 km/h. However, double the speed of the third runner added to the speed of the first results in 7 km/h. If thrice the speed of the first runner is added to the original speeds of the other two, the result is 12 km/h. Using the matrix method, find the original speed of each runner.
- Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^T AX = 0 \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \). \[ A = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}, \quad A = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix} \] If \( \det(\text{adj}(2(A + I))) = 2^{\alpha} 3^{\beta} 5^{\gamma} \), \( \alpha, \beta, \gamma \in \mathbb{N} \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is:
- If \( A = \begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix} \), then find the value of \( K \) if \( A^2 = 6A + K I_2 \), where \( I_2 \) is the identity matrix.
- Let \( A \) be a square matrix of order 3 such that \( \text{det}(A) = -2 \) and \( \text{det}(3 \cdot \text{adj}(-6 \cdot \text{adj}(3A))) = 2^{m+n} \cdot 3^{mn} \), where \( m > n \). Then \( 4m + 2n \) is equal to:
Questions Asked in GATE ES exam
Courage : Bravery :: Yearning :
Select the most appropriate option to complete the analogy.In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \( P \), \( Q \), and \( R \)?
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius \( r \) cm as shown in the figure. The side of the dodecagon is \( d \) cm. All the triangles (numbered 1 to 12 in the figure) are used to form squares of side \( r \) cm, and each numbered triangle is used only once to form a square. The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
The number of patients per shift (X) consulting Dr. Gita in her past 100 shifts is shown in the figure. If the amount she earns is ₹1000(X - 0.2), what is the average amount (in ₹) she has earned per shift in the past 100 shifts?
- Water from a hand pump located near a landfill has 1 mg/L arsenic (oral carcinogenic potency factor = 1.75 (kg-day)/mg). A person who lives nearby drinks 2 L/day of water from this hand pump for 10 years. Assume a body weight of 70 kg and an average life duration of 70 years. The chances of this person getting an excess risk of cancer is ________ × 10-3 (rounded off to three decimal places).
- GATE ES - 2025
- Hydrology