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Verify the statements in Exercises $19-24 .$ The matrices are square.If $A$ and $B$ are similar, then they have the same rank. [Hint: Refer to Supplementary Exercises 13 and 14 for Chapter 4.1

$\operatorname{rank} B P^{-1}=\operatorname{rank} B$ (Since $P^{-1}$ is invertible).And hence $\operatorname{rank} A=\operatorname{rank} B$Thus, if $A$ and $B$ are similar, then they have the same rank.

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 4

Eigenvectors and Linear Transformations

Vectors

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were asked to prove that if two matrices a and beer similar, then they have the same rank. So a and B are similar. This means that is equal to P B P in verse for some in verbal matrix p. So we have that the rank of A is equal to the rank and here I'm going to use parentheses for emphasis of p times be p in verse. Now we know by a previous the're, um that the rank of a product of matrices if the first Matrix is in vertebral is simply the rank of the second Matrix. So because P isn't variable, this is equal to the rank of be times P in verse. Likewise, we know that P inverse is also in vertebral and by another exercise. We know that the rank of a matrix times in verbal matrix is simply the rank of the first matrix. This is equal to the rank of be since PM verses also in vertebral

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