Register

TIFR-2015-Maths-A-3

• edited by
635 views
1 votes
1 votes

Let $A$ be a $10 \times 10$ matrix with complex entries such that all its eigenvalues are non-negative real numbers, and at least one eigenvalue is positive. Which of the following statements is always false ?

  1. There exists a matrix $B$ such that $AB-BA = B$
  2. There exists a matrix $B$ such that $AB-BA = A$
  3. There exists a matrix $B$ such that $AB+BA=A$
  4. There exists a matrix $B$ such that $AB+BA=B$
• edited by
User Avatar

1 Answer

0 votes
0 votes
Statement 1: There exists a matrix B such that AB + BA = A. This statement is true and can be satisfied for certain matrices A and B. For example, if A is the zero matrix, then any matrix B will satisfy AB + BA = A.

Statement 2: There exists a matrix B such that AB - BA = B. This statement is true and can be satisfied for certain matrices A and B. For example, let A be the zero matrix and B be any matrix. Then AB - BA = 0 - 0 = 0, which is equal to B.

Statement 3: There exists a matrix B such that AB - BA = A. This statement is always false. It is not possible to find a matrix B that satisfies AB - BA = A for all matrices A with non-negative real eigenvalues and at least one positive eigenvalue. The equation AB - BA = A can be rearranged as AB = BA + A, and taking the trace of both sides yields Tr(AB) = Tr(BA + A). However, the trace is cyclic, so Tr(AB) = Tr(BA), and the equation reduces to Tr(AB) = Tr(BA + A). Since the eigenvalues of A are non-negative real numbers, the trace of A is non-negative. However, the trace of BA + A may or may not be non-negative, making this equation impossible to satisfy for all A.

Statement 4: There exists a matrix B such that AB + BA = B. This statement is true and can be satisfied for certain matrices A and B. For example, let B be the zero matrix. Then AB + BA = A0 + 0A = 0, which is equal to B.

Therefore, the statement that is always false is Statement 3: There exists a matrix B such that AB - BA = A.
User Avatar
Voting & Accepting an answer helps improve the community
Answer:
← PreviousNext →
← Previous in categoryNext in category →

Related questions

5 votes
5 votes
2 answers
1
makhdoom ghaya asked Dec 19, 2015
697 views
TIFR-2015-Maths-A-6
Let $A$ be the $2 \times 2$ matrix $\begin{pmatrix} \sin\frac{\pi}{18}&-\sin \frac{4\pi}{9} \\ \sin \frac{4\pi}{9}&\sin \frac {\pi}{18} \end{pmatrix}$. Then the smallest number $n \in \mathbb{N}$ such that $A^{n}=1$ is. $3$ $9$ $18$ $27$
Let $A$ be the $2 \times 2$ matrix $\begin{pmatrix}\sin\frac{\pi}{18}&-\sin \frac{4\pi}{9} \\\sin \frac{4\pi}{9}&\sin \frac {\pi}{18}\end{pmatrix}$. Then the smallest num...
User Avatar
7 votes
7 votes
2 answers
2
makhdoom ghaya asked Dec 19, 2015
2,084 views
TIFR-2015-Maths-A-1
Let $A$ be an invertible $10 \times 10$ matrix with real entries such that the sum of each row is $1$. Then The sum of the entries of each row of the inverse of $A$ is $1$ The sum of the entries of each column of the inverse of $A$ is $1$ The trace of the inverse of $A$ is non-zero None of the above
Let $A$ be an invertible $10 \times 10$ matrix with real entries such that the sum of each row is $1$. ThenThe sum of the entries of each row of the inverse of $A$ is $1$...
User Avatar
5 votes
5 votes
2 answers
3
makhdoom ghaya asked Dec 20, 2015
958 views
TIFR-2015-Maths-B-5
Let $n \geq 1$ and let $A$ be an $n \times n$ matrix with real entries such that $A^{k}=0$, for some $k \geq 1$. Let $I$ be the identity $n \times n$ matrix. Then. $I+A$ need not be invertible. Det $(I+A)$ can be any non-zero real number. Det $(I+A) = 1$ $A^{n}$ is a non-zero matrix.
Let $n \geq 1$ and let $A$ be an $n \times n$ matrix with real entries such that $A^{k}=0$, for some $k \geq 1$. Let $I$ be the identity $n \times n$ matrix. Then.$I+A$ n...
User Avatar
4 votes
4 votes
1 answer
4
makhdoom ghaya asked Dec 9, 2015
1,136 views
TIFR-2011-Maths-A-3
Let $A$ be a $5 \times 5$ matrix with real entries, then $A$ has An eigenvalue which is purely imaginary At least one real eigenvalue At least two eigenvalues which are not real At least $2$ distinct real eigenvalues
Let $A$ be a $5 \times 5$ matrix with real entries, then $A$ hasAn eigenvalue which is purely imaginaryAt least one real eigenvalueAt least two eigenvalues which are not ...
User Avatar
Link Copied!