5dd4dd…: “On a navy square”

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True/False: Let $A$ be a square matrix. If $\dim \Nul A = 0$ then $A$ is invertible.

True False

True/False: Let $A$ be an $n\times n$ matrix. If $\rank A = n$ then $A$ is invertible.

True False

If $A$ is a $3\times 5$ matrix with a two-dimensional null space, what is the rank of $A$ ?

$\rank A = \answer {3}$

If $A$ is a $2\times 3$ matrix with $\dim \Col A = 3$ , what is $\dim \Nul A$ ?

$\dim \Nul A = \answer {0}$

If $A$ is a $8\times 7$ matrix with $\dim \Col A = 3$ , what is the rank of $A$ ?

$\rank A = \answer {3}$

$\rank A = \dim \Col A$

If $A$ is a $8\times 7$ matrix with $\dim \Col A = 3$ , what is the rank of $A$ ?

$\rank A = \answer {3}$

$\rank A = \dim \Col A$

$A$ and $B$ below are row equivalent. Determine the following values. $A = \begin {bmatrix} 1&-1& 2&3&1&0\\ 0&5&-2&1&1&1\\ -3&3&-6&-9&-3&0\\-1&1&2&1&-2&1\end {bmatrix} \hspace {10pt} B = \begin {bmatrix} 1&-1&2&3&1&0\\ 0&5&-2&1&1&1\\ 0&0&4&4&-1&1\\ 0&0&0&0&0&0\end {bmatrix}$

$\rank A = \answer {3}$
$\dim \Nul A = \answer {3}$

A basis for $\Col A$ is $\left \{ \begin {bmatrix} \answer {1}\\\answer {0}\\ \answer {-3}\\ \answer {-1}\end {bmatrix}, \begin {bmatrix} \answer {-1}\\\answer {5}\\ \answer {3}\\ \answer {1}\end {bmatrix}, \begin {bmatrix} \answer {2}\\\answer {-2}\\ \answer {-6}\\ \answer {2}\end {bmatrix} \right \}$