Problem #5: Consider the following five statements about similar matrices.(i) If A and B are similar matrices, then A2 and B2 are similar.(ii) If A and B are similar matrices, then at least one of A and B is a triangular matrix.Problem #5:(iii) If A and B are similar matrices and A is symmetric, then B is symmetric.(iv) If A and B are similar matrices, then det(A) = det(B).(v) If A and B are similar matrices, then A and B have the same eigenvalues.Determine which which statements are true (1) or false (2) by testing out each statement on an appropriate matrix.So, for example, if you think that the answers, in the above order, are True,False,False, True,False, then you wouldenter '1,2,2,1,2' into the answer box below (without the quotes).

Definition : A matrix A is said to be similar to another matrix B , if there exist a non…

Answered: Problem #5: the following five state…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Problem #5: Consider the following five statements about similar matrices.
(i) If A and B are similar matrices, then A2 and B2 are similar.
(ii) If A and B are similar matrices, then at least one of A and B is a triangular matrix.
Problem #5:
(iii) If A and B are similar matrices and A is symmetric, then B is symmetric.
(iv) If A and B are similar matrices, then det(A) = det(B).
(v) If A and B are similar matrices, then A and B have the same eigenvalues.
Determine which which statements are true (1) or false (2) by testing out each statement on an appropriate matrix.
So, for example, if you think that the answers, in the above order, are True,False,False, True,False, then you would
enter '1,2,2,1,2' into the answer box below (without the quotes).

Expert Solution

Step 1: Part (i) and (ii)

Definition : A $n cross times n$ matrix A is said to be similar to another $n cross times n$ matrix B , if there exist a non singular matrix P such that

$B equals P to the power of negative 1 end exponent A P$

( ii ) If A and B are similar matrices, then at least one of A and B is a triangular matrix .

Answer : False

Explanation : $A equals open square brackets table row cell negative 1 end cell 2 row 3 1 end table close square brackets$ and $B equals space open square brackets table row cell negative 5 end cell cell negative 3 end cell row 6 5 end table close square brackets$ then A is similar to B . Because there exist a non singular matrix P such that $P equals open square brackets table row 2 1 row cell negative 1 end cell 0 end table close square brackets$ $rightwards double arrow space$ $P to the power of negative 1 end exponent equals space open square brackets table row 0 cell negative 1 end cell row 1 2 end table close square brackets$ such that $P to the power of negative 1 end exponent A P space equals space B$ .

Here A and B none of these triangular matrices.

(i) If A and B are similar matrices, then $A squared$ and $B squared$ are also similar

Answer : True

Explanation : $A space tilde space B$ $rightwards double arrow space$ $P to the power of negative 1 end exponent A P space equals space B$

$rightwards double arrow space$ $B squared space equals space B. B space equals space open parentheses P to the power of negative 1 end exponent A P close parentheses space open parentheses P to the power of negative 1 end exponent A P close parentheses equals P to the power of negative 1 end exponent A I A P space equals space P to the power of negative 1 end exponent A squared P$

$rightwards double arrow space$ $A squared space tilde space B squared$