Consider the following five statements about similar matrices.(i) If A and B are similar matrices, then at least one of A and B is a triangular matrix.(ii) If A and B are similar matrices, then det(A) = det(B).(iii) If A and B are similar matrices, then A² and B² are similar.(iv) If A and B are similar matrices, then A and B have the same eigenvalues.(v) If A and B are similar matrices and A is symmetric, then B is symmetric.Determine which which statements are true (1) or false (2) by testing out each statement on an appropriatematrix.

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

Answered: Consider the following five statements…

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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Consider the following five statements about similar matrices.
(i) If A and B are similar matrices, then at least one of A and B is a triangular matrix.
(ii) If A and B are similar matrices, then det(A) = det(B).
(iii) If A and B are similar matrices, then A² and B² are similar.
(iv) If A and B are similar matrices, then A and B have the same eigenvalues.
(v) If A and B are similar matrices and A is symmetric, then B is symmetric.
Determine which which statements are true (1) or false (2) by testing out each statement on an appropriate
matrix.

Expert Solution

Step 1: Part (i) and (iii)

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

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

( i ) 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.

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

Answer : True

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

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

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