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Chapter 5: Q5.4-24E (page 267)

Verify the statements in Exercises 19–24. The matrices are square.

24. If \(A\) and \(B\) are similar, then they have the same rank. (Hint: Refer to supplementary Exercises 13 and 14 for Chapter 4.)

Short Answer

Expert verified

It is proved that \(A\) and \(B\) have the same rank.

Step by step solution

01

Rank of the matrix

It is known that when \(P\) is aninvertible \(m \times m\) matrix, then \({\mathop{\rm rank}\nolimits} A = {\mathop{\rm rank}\nolimits} B\) .

Also, when \(Q\) is invertible, then \({\mathop{\rm rank}\nolimits} AQ = {\mathop{\rm rank}\nolimits} A\).

02

Show that \(A\) and \(B\) have the same rank 

When \(A = PB{P^{ - 1}}\) then \({\mathop{\rm rank}\nolimits} A = {\mathop{\rm rank}\nolimits} P\left( {B{P^{ - 1}}} \right) = {\mathop{\rm rank}\nolimits} B{P^{ - 1}}\) , according to the above statement. Moreover, \({\mathop{\rm rank}\nolimits} B{P^{ - 1}} = {\mathop{\rm rank}\nolimits} B\), according to the above statement, because \({P^{ - 1}}\) is invertible.

Therefore, \({\mathop{\rm rank}\nolimits} A = {\mathop{\rm rank}\nolimits} B\) .

Thus, it is proved that \(A\) and \(B\) have the same rank.

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Most popular questions from this chapter

Question: Construct a random integer-valued \(4 \times 4\) matrix \(A\).

  1. Reduce \(A\) to echelon form \(U\) with no row scaling, and use \(U\) in formula (1) (before Example 2) to compute \(\det A\). (If \(A\) happens to be singular, start over with a new random matrix.)
  2. Compute the eigenvalues of \(A\) and the product of these eigenvalues (as accurately as possible).
  3. List the matrix \(A\), and, to four decimal places, list the pivots in \(U\) and the eigenvalues of \(A\). Compute \(\det A\) with your matrix program, and compare it with the products you found in (a) and (b).

Let \(A{\bf{ = }}\left( {\begin{aligned}{*{20}{c}}{{a_{{\bf{11}}}}}&{{a_{{\bf{12}}}}}\\{{a_{{\bf{21}}}}}&{{a_{{\bf{22}}}}}\end{aligned}} \right)\). Recall from Exercise \({\bf{25}}\) in Section \({\bf{5}}{\bf{.4}}\) that \({\rm{tr}}\;A\) (the trace of \(A\)) is the sum of the diagonal entries in \(A\). Show that the characteristic polynomial of \(A\) is \({\lambda ^2} - \left( {{\rm{tr}}A} \right)\lambda + \det A\). Then show that the eigenvalues of a \({\bf{2 \times 2}}\) matrix \(A\) are both real if and only if \(\det A \le {\left( {\frac{{{\rm{tr}}A}}{2}} \right)^2}\).

Question: Construct a random integer-valued \(4 \times 4\) matrix \(A\), and verify that \(A\) and \({A^T}\) have the same characteristic polynomial (the same eigenvalues with the same multiplicities). Do \(A\) and \({A^T}\) have the same eigenvectors? Make the same analysis of a \(5 \times 5\) matrix. Report the matrices and your conclusions.

Question: Find the characteristic polynomial and the eigenvalues of the matrices in Exercises 1-8.

2. \(\left[ {\begin{array}{*{20}{c}}5&3\\3&5\end{array}} \right]\)

a. Let \(A\) be a diagonalizable \(n \times n\) matrix. Show that if the multiplicity of an eigenvalue \(\lambda \) is \(n\), then \(A = \lambda I\).

b. Use part (a) to show that the matrix \(A =\left({\begin{aligned}{*{20}{l}}3&1\\0&3\end{aligned}}\right)\) is not diagonalizable.
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