Question:
If the matrices \(\left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right),\left(\begin{matrix} -1 & 0 \\ 0 & 0 \end{matrix}\right),\left(\begin{matrix} i & 0 \\ 0 & 0 \end{matrix}\right) and \left(\begin{matrix} -i & 0 \\ 0 & 0 \end{matrix}\right)\) form a group with respect to matrix multiplication, then which one of the following statements about the groups, thus formed is correct?
If the matrices \(\left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right),\left(\begin{matrix} -1 & 0 \\ 0 & 0 \end{matrix}\right),\left(\begin{matrix} i & 0 \\ 0 & 0 \end{matrix}\right) and \left(\begin{matrix} -i & 0 \\ 0 & 0 \end{matrix}\right)\) form a group with respect to matrix multiplication, then which one of the following statements about the groups, thus formed is correct?
Updated On: Mar 12, 2025
- The group has no element of order 4.
- The group has an element of order 3.
- The group is non-abelian
- \(\left(\begin{matrix} -1 & 0 \\ 0 & 0 \end{matrix}\right)\)is its own inverse
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
The correct answer is(D): \(\left(\begin{matrix} -1 & 0 \\ 0 & 0 \end{matrix}\right)\)is its own inverse
Was this answer helpful?
0
0
Top Questions on Matrices
- Four friends Abhay, Bina, Chhaya, and Devesh were asked to simplify \( 4 AB + 3(AB + BA) - 4 BA \), where \( A \) and \( B \) are both matrices of order \( 2 \times 2 \). It is known that \( A \neq B \) and \( A^{-1} \neq B \). Their answers are given as:
- Given \[ A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix} \] find \( AB \). Hence, solve the system of linear equations: \[ x - y + z = 4, \] \[ x - 2y - 2z = 9, \] \[ 2x + y + 3z = 1. \]
- If \[ A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix} \] then find \( A^{-1} \). Hence, solve the system of linear equations: \[ x - 2y = 10, \] \[ 2x - y - z = 8, \] \[ -2y + z = 7. \]
- Assertion (A): \( A = \text{diag} [3, 5, 2] \) is a scalar matrix of order \( 3 \times 3 \).
Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix. - If \( A \) and \( B \) are square matrices of order \( m \) such that \( A^2 - B^2 = (A - B)(A + B) \), then which of the following is always correct?
View More Questions
Questions Asked in CUET PG exam
- What is the correct verb form: He, as well as his friends, ?
- Which preposition is correct: Chanakya lived ?
- CUET (PG) - 2025
- Prepositions
- Identify the possessive noun: My office car is parked elsewhere.
- Which tense is appropriate: How did Saira ?
- Complete the sentence: There is a growing synthesis between humanism and ?
- CUET (PG) - 2025
- Sentence Completion
View More Questions