0c6949…: “Instead of the fall number”

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True/False: $n\times n$ matrices $A$ and $B$ are said to be similar if $A = PBP^{-1}$ for some invertible matrix $P$ .

True False

Let $D = \begin {bmatrix} 2 &0\\ 0&5\end {bmatrix}$ and let $k\geq 1$ . What is $D^k$ ?

$D^k = \begin {bmatrix} 2^k &0\\ 0&5^k\end {bmatrix}$ $D^k = \begin {bmatrix} 2k &0\\ 0&5k\end {bmatrix}$ $D^k = \begin {bmatrix} 2 &k\\ k&5\end {bmatrix}$ $D^k = \begin {bmatrix} 2^k &k\\ k&5^k\end {bmatrix}$

Let $D = \begin {bmatrix} 2 &0\\ 0&5\end {bmatrix}$ and let $k\geq 1$ . What is $D^k$ ?

Let $A = PDP^{-1}$ such that $P= \begin {bmatrix} 3 &-1 \\ -8 &3\end {bmatrix}$ and $D = \begin {bmatrix} 2 &0 \\ 0 &-2\end {bmatrix}$ . Compute the following:

$A^4 = \begin {bmatrix} \answer {16} &\answer {0}\\\answer {0} &\answer {16}\end {bmatrix}$

$A^3 = \begin {bmatrix} \answer {136} &\answer {48}\\\answer {-384} &\answer {-136}\end {bmatrix}$

True/False: An $n\times n$ matrix $A$ is diagonalizable if and only if $A$ has exactly $n$ eigenvectors.

True False

Read the Diagonalization Theorem.

True/False: If a $4\times 4$ matrix $A$ has a linearly independent set of four eigenvectors, then $A$ is diagonalizable.

True False

Read the Diagonalization Theorem.

True/False: It is possible for an $n\times n$ matrix $A$ to have a linearly independent set of more than $n$ eigenvectors.

True False

$A$ has $n$ eigenvalues (counting multiplicities). For each eigenvalue, the eigenspace has dimension less than or equal to the multiplicity of the eigenvalue.