$A=\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$

atgnybo4fq
2022-11-12
Answered

Determining the Rank of a Matrix

$A=\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$

$A=\left(\begin{array}{ccc}1& 2& 4\\ 3& 6& 5\end{array}\right)$

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Deanna Sweeney

Answered 2022-11-13
Author has **14** answers

The matrix has $2$ rows and $3$ columns, so the greatest possible value of its rank is $2$. We pick any element which is not $0$.

$\left(\begin{array}{ccc}{1}& 2& 4\\ 3& 6& 5\end{array}\right)$

We form an order $2$ minor containing $1$.

$\left(\begin{array}{ccc}{1}& {2}& 4\\ {3}& {6}& 5\end{array}\right)$

We calculate this minor.

$\left[\begin{array}{cc}{1}& {2}\\ {3}& {6}\end{array}\right]=6-6=0$

We form another order $3$ minor containing $1$. $A=\left(\begin{array}{ccc}{1}& 2& {4}\\ {3}& 6& {5}\end{array}\right)$

We calculate this minor.

$\left[\begin{array}{cc}{1}& {4}\\ {3}& {5}\end{array}\right]=5-12=-7\ne 0$

The rank is $2$

$\left(\begin{array}{ccc}{1}& 2& 4\\ 3& 6& 5\end{array}\right)$

We form an order $2$ minor containing $1$.

$\left(\begin{array}{ccc}{1}& {2}& 4\\ {3}& {6}& 5\end{array}\right)$

We calculate this minor.

$\left[\begin{array}{cc}{1}& {2}\\ {3}& {6}\end{array}\right]=6-6=0$

We form another order $3$ minor containing $1$. $A=\left(\begin{array}{ccc}{1}& 2& {4}\\ {3}& 6& {5}\end{array}\right)$

We calculate this minor.

$\left[\begin{array}{cc}{1}& {4}\\ {3}& {5}\end{array}\right]=5-12=-7\ne 0$

The rank is $2$

asked 2021-09-13

Suppose that A is row equivalent to B. Find bases for the null space of A and the column space of A.

$A=\left[\begin{array}{ccccc}1& 2& -5& 11& -3\\ 2& 4& -5& 15& 2\\ 1& 2& 0& 4& 5\\ 3& 6& -5& 19& -2\end{array}\right]$

$B=\left[\begin{array}{ccccc}1& 2& 0& 4& 5\\ 0& 0& 5& -7& 8\\ 0& 0& 0& 0& -9\\ 0& 0& 0& 0& 0\end{array}\right]$

asked 2021-06-13

For the matrix A below, find a nonzero vector in the null space of A and a nonzero vector in the column space of A

$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$

Find a vector in the null space of A that is not the zero vector

$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

asked 2021-09-18

I need to find a unique description of Nul A, namely by listing the vectors that measure the null space.

$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

asked 2021-09-15

T must be a linear transformation, we assume. Can u find the T standard matrix.$T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{4},T\left({e}_{1}\right)=(3,1,3,1)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,2,0,0),\text{}where\text{}{e}_{1}=(1,0)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{e}_{2}=(0,1)$.

asked 2022-01-30

Find invariant points under matrix transformation

The matrix:

$Q=\left[\begin{array}{cc}-1& 2\\ 0& 1\end{array}\right]$

The matrix:

asked 2021-12-26

Assume that T is a linear transformation. Find the standard matrix of T.

T:${\mathbb{R}}^{2}\to {\mathbb{R}}^{4},\text{}T\left({e}_{1}\right)=(3,1,3,1),\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,6,0,0)$ , where ${e}_{1}=(1,0)$ and ${e}_{2}=(0,1)$

$A=$

T:

asked 2022-07-24

a5=0 and a15=4 what is the sum of the first 15 terms of that arithmetic sequence