Question:
Let $A$ be a matrix such that $A^2 = I$, where $I$ is an identity matrix. Then $(I + A)^4 - 8A$ is equal to:
Let $A$ be a matrix such that $A^2 = I$, where $I$ is an identity matrix. Then $(I + A)^4 - 8A$ is equal to:
Updated On: Nov 28, 2024
$8 I$
$5 I$
- 8(I + A)
- 5(I - A)
Hide Solution
Verified By Collegedunia
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: \[8I\]
Was this answer helpful?
0
0
Top Questions on Matrices
- If \(A = \begin{bmatrix} 3 & 2 \\ -1 & 1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -1 & 0 \\ 2 & 5 \\ 3 & 4 \end{bmatrix},\)then \((BA)^T\) is equal to:
- If \( A = \begin{bmatrix} K & 4 \\ 4 & K \end{bmatrix} \) and \( |A^3| = 729 \), then the value of \( K^8 \) is:
- Let $A = I_2 - MM^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M = I_1$ holds. If $\lambda$ is a real number such that the relation $AX = \lambda X$ holds for some non-zero real matrix $X$ of order $2 \times 1$, then the sum of squares of all possible values of $\lambda$ is equal to:
- Let the system of equations \[x + 2y + 3z = 5, \quad 2x + 3y + z = 9, \quad 4x + 3y + \lambda z = \mu\]have an infinite number of solutions. Then $\lambda + 2\mu$ is equal to:
- For \( \alpha, \beta \in \mathbb{R} \) and a natural number \( n \), let \[A_r = \begin{vmatrix} r & 1 & \frac{n^2}{2} + \alpha \\ 2r & 2 & n^2 - \beta \\3r - 2 & 3 & \frac{n(3n - 1)}{2} \end{vmatrix}.\]Then \( 2A_{10} - A_8 \) is:
View More Questions
Questions Asked in CUET exam
- A molecule X associates in a given solvent as per the following equation:
X ⇌ (X)n
For a given concentration of X, the van’t Hoff factor was found to be 0.80 and the
fraction of associated molecules was 0.3. The correct value of ‘n’ is:- CUET (UG) - 2024
- Solutions
- Who devised the concept of Intelligence Quotient (IQ)?
- CUET (UG) - 2024
- Psychological attributes
- Two pipes A and B can fill a tank in 32 minutes and 48 minutes respectively. If both the pipes are opened simultaneously, after how much time B should be turned off so that the tank is full in 20 minutes?
- CUET (UG) - 2024
- Time and Work
- The spin-only magnetic moment of Hexacyanomanganate(II) ion is.........BM
- CUET (UG) - 2024
- Magnetic moment
- Match List-I with List-II.
List-I List-II (A) Confidence level (I) Percentage of all possible samples that can be expected to include the true population parameter (B) Significance level (III) The probability of making a wrong decision when the null hypothesis is true (C) Confidence interval (II) Range that could be expected to contain the population parameter of interest (D) Standard error (IV) The standard deviation of the sampling distribution of a statistic Choose the correct answer from the options given below:- CUET (UG) - 2024
- Percentage
View More Questions