The eigenvalues of matrix
are 5 and 10. For matrix \( B = A + \alpha I \), where \( \alpha \) is a constant and \( I \) is the \( 2 \times 2 \) identity matrix, its eigenvalues are
The eigenvalues of matrix
are 5 and 10. For matrix \( B = A + \alpha I \), where \( \alpha \) is a constant and \( I \) is the \( 2 \times 2 \) identity matrix, its eigenvalues are
Show Hint
- 5, 10
- \( 5 + \alpha, 10 + \alpha \)
- \( 5 - \alpha, 10 - \alpha \)
- \( 5\alpha, 10\alpha \)
The Correct Option is B
Solution and Explanation
We are given that the eigenvalues of matrix 
are 5 and 10. Now, we are asked to find the eigenvalues of matrix \( B \), which is given by \( B = A + \alpha I \), where \( I \) is the identity matrix. Step 1: Eigenvalues of matrix \( A \). The eigenvalues of matrix \( A \) are given as 5 and 10. We can express this as: \[ \text{Eigenvalues of } A: \lambda_1 = 5, \lambda_2 = 10. \] Step 2: Effect of adding \( \alpha I \). When a constant multiple of the identity matrix \( \alpha I \) is added to a matrix, the eigenvalues of the matrix are shifted by the constant \( \alpha \). Therefore, the eigenvalues of matrix \( B \) will be: \[ \lambda_B = \lambda_A + \alpha. \] Thus, the eigenvalues of matrix \( B \) are: \[ \lambda_{B1} = 5 + \alpha, \quad \lambda_{B2} = 10 + \alpha. \] Therefore, the correct answer is (B). Final Answer: (B) \( 5 + \alpha, 10 + \alpha \)
Top Questions on Eigenvalues
- The eigenvalues of the matrix \[ A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix} \] are \(\lambda_1, \lambda_2, \lambda_3\). The value of \(\lambda_1 \lambda_2 \lambda_3 (\lambda_1 + \lambda_2 + \lambda_3)\) is:
- The eigenvalues of the matrix \( A = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \) are:
The eigenvalues of the matrix
are \( \lambda_1, \lambda_2, \lambda_3 \). The value of \( \lambda_1 \lambda_2 \lambda_3 ( \lambda_1 + \lambda_2 + \lambda_3 ) \) is:
- The eigenvalues of the matrix: \[ \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \] are ________
Let \[ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix}. \] If the eigenvalues of \( A \) are -2, 1, and 2, then the value of \( k \) is _.
(Answer in integer)
Questions Asked in GATE PI 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 \( R \)?
- In the iron-carbon equilibrium phase diagram, the eutectoid reaction occurs at 723°C with the eutectoid composition of 0.83 weight % carbon. Ferrite and cementite phases are considered to contain 0.022 weight % carbon and 6.67 weight % carbon, respectively. If a steel specimen with 0.7 weight % carbon is cooled from 950°C to below 723°C, the fraction of eutectoid ferrite is .............
- GATE PI - 2025
- Static Equilibrium
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?
