Question

Identify whether the following statements are true or false. If a statement is false, give an explanation why. (a) An $n \times n$ matrix has $n$ eigenvalues. (b) $\frac{6 - 12i}{2 + 3i} = 3 - 4i$ (c) If A is similar to the identity matrix, then det $A = 0$. (d) If $x$ is an eigenvector of A, then the line through the origin and $x$ passes through $Ax$. (e) If B and C are bases for the same vector space V, then B and C contain the same number of vectors. (f) If $u$ and $v$ are in $\mathbb{R}^n$, then $u \cdot v = v \cdot u$. (g) A is a diagonalizable matrix if $A = PDP^{-1}$ for some matrix D and some invertible matrix P.

          Identify whether the following statements are true or false. If a statement is false, give an explanation why.
(a) An $n \times n$ matrix has $n$ eigenvalues.
(b) $\frac{6 - 12i}{2 + 3i} = 3 - 4i$
(c) If A is similar to the identity matrix, then det $A = 0$.
(d) If $x$ is an eigenvector of A, then the line through the origin and $x$ passes through $Ax$.
(e) If B and C are bases for the same vector space V, then B and C contain the same number of vectors.
(f) If $u$ and $v$ are in $\mathbb{R}^n$, then $u \cdot v = v \cdot u$.
(g) A is a diagonalizable matrix if $A = PDP^{-1}$ for some matrix D and some invertible matrix P.

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Identify whether the following statements are true or false. If a statement is false, give an explanation why.
(a) An n × n matrix has n eigenvalues.
(b) (6 - 12i)/(2 + 3i) = 3 - 4i
(c) If A is similar to the identity matrix, then det A = 0.
(d) If x is an eigenvector of A, then the line through the origin and x passes through Ax.
(e) If B and C are bases for the same vector space V, then B and C contain the same number of vectors.
(f) If u and v are in ℝ^n, then u · v = v · u.
(g) A is a diagonalizable matrix if A = PDP^-1 for some matrix D and some invertible matrix P.

Added by Debbie M.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Adi S

06/21/2023

Step 1

This is not necessarily true. While an n x n matrix has n eigenvalues, they may not all be distinct. Some eigenvalues may have multiplicity greater than 1, meaning they are repeated. Therefore, the statement is false. Show more…

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Identify whether the following statements are true or false. If a statement is false, give an explanation why. (a) An n imes n matrix has n eigenvalues. (b) (6-12i)/(2+3i)=3-4i (c) If A is similar to the identity matrix, then detA=0. (d) - If x is an eigenvector of A, then the line through the origin and x passes through Ax. (e) If B and C are bases for the same vector space V, then B and C contain the same number of vectors. (f) If u and v are in R^(n), then u*v=v*u. (g) A is a diagonalizable matrix if A=PDP^(-1) for some matrix D and some invertible matrix P. Identify whether the following statements are true or false. If a statement is false, give an explanation why (a) An n x n matrix has n eigenvalues. 6-12i (b) =3-4i 2+3i (c) If A is similar to the identity matrix, then det A = 0. (d)-If x is an eigenvector of A, then the line through the origin and x passes through Ax (e) If B and C are bases for the same vector space V, then B and C contain the same number of vectors (f) If u and v are in Rn, then u.v =v.u. matrix P. that

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