Problem 15
Question
Let \(\mathrm{A}\), other than \(\mathrm{I}\) or \(-\mathrm{I}\), be a \(2 \times 2\) real matrix such that \(\mathrm{A}^{2}=\mathrm{I}, \mathrm{I}\) being the unit matrix. Let \(\operatorname{Tr}(\mathrm{A})\) be the sum of diagonal elements of A. \(\quad\) [Online April 23, 2013] Statement-1: \(\operatorname{Tr}(\mathrm{A})=0\) Statement-2: \(\operatorname{det}(\mathrm{A})=-1\) (a) Statement- 1 is true; Statement- 2 is false. (b) Statement- 1 is true; Statement- 2 is true; Statement-2 is not a correct explanation for Statement-1. (c) Statement- 1 is true; Statement- 2 is true; Statement-2 is a correct explanation for Statement- 1 . (d) Statement- 1 is false; Statement- 2 is true.
Step-by-Step Solution
Verified Answer
(b) Statement- 1 is true; Statement- 2 is true; Statement-2 is not a correct explanation for Statement-1.
1Step 1: Understand the Problem
We are given a matrix \( A \) such that \( A^2 = I \), where \( I \) is the identity matrix. We need to analyze two statements regarding the trace and determinant of \( A \) and determine their validity and relation.
2Step 2: Analyze Matrix Condition
Given \( A^2 = I \), this implies that \( A \) is an involutory matrix, which means every eigenvalue \( \lambda \) of \( A \) satisfies \( \lambda^2 = 1 \). Therefore, the eigenvalues of \( A \) must be either 1 or -1.
3Step 3: Consider the Trace of A
The trace of a matrix \( \operatorname{Tr}(A) \) is the sum of its eigenvalues. Since the eigenvalues of \( A \) are either 1 or -1, \( A \) having one eigenvalue of 1 and another of -1 would make \( \operatorname{Tr}(A) = 1 + (-1) = 0 \). Thus, Statement-1 is true.
4Step 4: Consider the Determinant of A
The determinant of a matrix is the product of its eigenvalues. For matrix \( A \), having eigenvalues 1 and -1, the determinant is \( \,1 \times (-1) = -1 \, \). Thus, Statement-2 is also true.
5Step 5: Examine Logical Connection
While both statements are true, the determinant being -1 does not directly explain why the trace is zero. The zero trace comes from the opposing values of the eigenvalues adding to zero, rather than their product influencing it.
6Step 6: Conclusion
Based on the analysis above, both Statement-1 is true and Statement-2 is true; however, Statement-2 is not a correct explanation for Statement-1.
Key Concepts
Involutory MatrixEigenvaluesDeterminantTrace of a Matrix
Involutory Matrix
An involutory matrix, such as the matrix \( A \) from our exercise, holds the unique property where squaring the matrix yields the identity matrix: \( A^2 = I \). This special type of matrix is called involutory because it essentially "undoes" itself when applied twice. Imagine it like pressing a reset button. You end up back at square one, or the identity matrix, in this case.
A key characteristic of an involutory matrix is evident when examining its eigenvalues. Since \( A^2 = I \), it means each eigenvalue \( \lambda \) must satisfy \( \lambda^2 = 1 \). Recall that \( \lambda^2 - 1 = 0 \) can be factored into \( (\lambda - 1)(\lambda + 1) = 0 \). Therefore, possible eigenvalues are \( 1 \) and \( -1 \).
This property is critical in simplifying many matrix-related problems, allowing us to quickly deduce features like trace and determinant related to eigenvalues.
A key characteristic of an involutory matrix is evident when examining its eigenvalues. Since \( A^2 = I \), it means each eigenvalue \( \lambda \) must satisfy \( \lambda^2 = 1 \). Recall that \( \lambda^2 - 1 = 0 \) can be factored into \( (\lambda - 1)(\lambda + 1) = 0 \). Therefore, possible eigenvalues are \( 1 \) and \( -1 \).
This property is critical in simplifying many matrix-related problems, allowing us to quickly deduce features like trace and determinant related to eigenvalues.
Eigenvalues
Eigenvalues are essential in matrix theory as they tell us about the matrix's fundamental properties. For any matrix \( A \), the eigenvalues are determined by solving the characteristic equation, which in this context leads to values where \( A - \lambda I \) has no inverse. In our problem, the matrix \( A \) is involutory with eigenvalues of either \( 1 \) or \( -1 \).
The eigenvalues essentially indicate how the matrix "stretches" spaces. With eigenvalues of \( 1 \), vectors remain unchanged in magnitude, whereas with \( -1 \), they reverse direction. Therefore, for the involutory matrix \( A \), the presence of both eigenvalues suggests that certain vectors don't change direction while others do. This interplay is key in determining the matrix characteristics like trace and determinant.
The eigenvalues essentially indicate how the matrix "stretches" spaces. With eigenvalues of \( 1 \), vectors remain unchanged in magnitude, whereas with \( -1 \), they reverse direction. Therefore, for the involutory matrix \( A \), the presence of both eigenvalues suggests that certain vectors don't change direction while others do. This interplay is key in determining the matrix characteristics like trace and determinant.
Determinant
The determinant of a matrix is a scalar that tells us about its fundamental properties, especially concerning invertibility and volume scaling. For a 2x2 matrix \( A \) with eigenvalues \( 1 \) and \( -1 \), the determinant is the product: \( 1 \cdot (-1) = -1 \).
In our problem, the determinant reflects the matrix's ability to reverse orientation in two-dimensional space due to its negative value. This result signals that the matrix transforms space in such a way that it includes a mirroring effect. Additionally, a determinant of \(-1\) verifies \( A \) as non-singular, affirming its invertibility.
In our problem, the determinant reflects the matrix's ability to reverse orientation in two-dimensional space due to its negative value. This result signals that the matrix transforms space in such a way that it includes a mirroring effect. Additionally, a determinant of \(-1\) verifies \( A \) as non-singular, affirming its invertibility.
Trace of a Matrix
The trace of a matrix, denoted as \( \operatorname{Tr}(A) \), is the sum of its main diagonal elements, or equivalently, the sum of its eigenvalues. For our matrix \( A \), possessing eigenvalues \( 1 \) and \( -1 \), this sum is clearly \( 1 + (-1) = 0 \).
The trace provides us quick insights into the matrix's eigenvalue composition. In many mathematical contexts, like our exercise involving an involutory matrix, knowing the trace helps determine potential properties without fully solving for the eigenvalues individually. In this case, a zero trace indicates an even balance of eigenvalues around zero, a hallmark of involutory matrices with properties like transformation symmetry in vector spaces.
The trace provides us quick insights into the matrix's eigenvalue composition. In many mathematical contexts, like our exercise involving an involutory matrix, knowing the trace helps determine potential properties without fully solving for the eigenvalues individually. In this case, a zero trace indicates an even balance of eigenvalues around zero, a hallmark of involutory matrices with properties like transformation symmetry in vector spaces.
Other exercises in this chapter
Problem 13
If \(\mathrm{B}\) is a \(3 \times 3\) matrix such that \(\mathrm{B}^{2}=0\), then det. \(\left[(\mathrm{I}+\mathrm{B})^{50}-50 \mathrm{~B}\right]\) is equal to:
View solution Problem 14
Let \(S=\left\\{\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right): a_{i j} \in\\{0,1,2\\}, a_{11}=a_{22}\right\\}\) Then the number of
View solution Problem 16
Statement \(-1:\) Determinant of a skew-symmetric matrix of order 3 is zero. Statement - 2 : For any matrix \(\mathrm{A}, \operatorname{det}(\mathrm{A})^{\mathr
View solution Problem 17
Let \(A\) be a \(2 \times 2\) matrix with non-zero entries and let \(A^{2}=I\), where \(I\) is \(2 \times 2\) identity matrix. Define \(\operatorname{Tr}(A)=\)
View solution