Question

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

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

Step-by-step solution

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 \({\mathop{\rm rank}\nolimits} AQ = {\mathop{\rm rank}\nolimits} A\).

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.