Let $A$ be a matrix such that $A^2 = I$, where $I$ is an identity matrix. Then $(I + A)^4 - 8A$ is equal to:
$8 I$
$5 I$
- 8(I + A)
- 5(I - A)
The Correct Option is A
Solution and Explanation
Given \(A^2 = I\), the identity matrix, we analyze \((I + A)^4\):
\[(I + A)^2 = I^2 + 2IA + A^2 = I + 2A + I = 2I + 2A\]
\[(I + A)^4 = ((I + A)^2)^2 = (2I + 2A)^2\]
Expanding \((2I + 2A)^2\):
\[(2I + 2A)^2 = 4I^2 + 8IA + 4A^2\]
Since \(A^2 = I\), substitute \(4A^2 = 4I\):
\[(2I + 2A)^2 = 4I + 8A + 4I = 8I + 8A\]
Now compute \((I + A)^4 - 8A\):
\[(I + A)^4 - 8A = (8I + 8A) - 8A = 8I\]
Thus, the result is: \[8I\]
Top Questions on Matrices
Let $A = \begin{bmatrix} \cos \theta & 0 & -\sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \cos \theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$, $A^2 = A^T$, then the sum of the diagonal elements of the matrix $(A + I)^3 + (A - I)^3 - 6A$ is equal to
- If \(A\) is a square matrix such that \(A^2 = A\), then \((I - A)^3\) is:
Let $ A $ be a $ 3 \times 3 $ matrix such that $ | \text{adj} (\text{adj} A) | = 81.
$ If $ S = \left\{ n \in \mathbb{Z}: \left| \text{adj} (\text{adj} A) \right|^{\frac{(n - 1)^2}{2}} = |A|^{(3n^2 - 5n - 4)} \right\}, $ then the value of $ \sum_{n \in S} |A| (n^2 + n) $ is:- Let $ A $ be a matrix of order $ 3 \times 3 $ and $ |A| = 5 $. If $ |2 \, \text{adj}(3A \, \text{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}, \quad \alpha, \beta, \gamma \in \mathbb{N} $ then $ \alpha + \beta + \gamma $ is equal to
- If \( A \) is a square matrix of order \( 3 \times 3 \), \( \det A = 3 \), then the value of \( \det(3A^{-1}) \) is:
Questions Asked in CUET exam
- A shopkeeper earned a profit (in Rs.) by selling an item, which is three times the discount offered (in Rs.). If the discount offered is 6.25%, what is his profit percentage?
- CUET (UG) - 2024
- Profit and Loss
- If 75$\%$ of a first-order reaction gets completed in 32 minutes, the time taken for 50$\%$ completion of this reaction is:
- CUET (UG) - 2024
- Order of reaction
- Two charged particles, placed at a distance d apart in vacuum, exert a force F on each other. Now, each of the charges is doubled. To keep the force unchanged, the distance between the charges should be changed to_____
Fill in the blank with the correct answer from the options given below.- CUET (UG) - 2024
- Electrostatics
- The total number of sigma bonds in P$_4$O$_{10}$ is:
- CUET (UG) - 2024
- Chemical bonding and molecular structure
- For the reaction 2A₂(g) + ¼X(g) → 2A₂X(g),If the volume is increased to double its value by decreasing the pressure on it. If the reaction is first order with respect to X and second order with respect to A2, the rate of reaction will:
- CUET (UG) - 2024
- Chemical Reaction