All Questions
Tagged with gaussian-elimination matrix-rank
17 questions
2
votes
0
answers
45
views
Rank of this matrix with a parameter: explanation about losing information
Consider the following matrix, where $k$ is a real parameter:
$$\begin{pmatrix} 1 & k & 1 \\ k & 1 & 1 \\ 1 & 1 & k \end{pmatrix}$$
I know I can study the zeroes of the ...
- 3,521
1
vote
1
answer
113
views
Determine rank of $ A = \begin{bmatrix} 2 & 1 & -2 & 1 \\ 4 & 1 & -2 & -3 \\ 1 & -1 & 2 & -3 \\ 2 & 2 & -4 & -5 \\ 3 & 1 & -2 & 2 \end{bmatrix}$
Could you give me your feedback ? I've verified with https://matrix.reshish.com/rankCalculation.php but maybe there are things that could be done differently
Determine the rank of the following ...
- 1,135
0
votes
0
answers
187
views
Does a (square) full rank matrix have identical column and row spaces?
Say we have a full rank matrix A (i.e., the rows are linearly independent and the same is true for columns). Since using the basic procedures of swapping, scalar multiplication, and addition, we can ...
0
votes
1
answer
378
views
Rank of a matrix if one of the diagonal elements is $0$ during elementary row operations
I'm required to find the rank of a square matrix A of dimension $m * m$. While doing elementary row operations I encountered a $0$ at a diagonal position. I tried all the possible row exchanges, but ...
- 169
1
vote
1
answer
69
views
Rank of a matrix using colunm space
I'd like some verification of my solution method since I don't have any solution at hand. I'm given the following matrix and asked to determine the rank of it in function of $x$:
$\begin{pmatrix}
2 &...
- 181
1
vote
2
answers
473
views
Making a matrix have a certain rank
Find the values of $a$ and $b$ such that
$$\mbox{rank} \begin{pmatrix}
3 & 2 & 5 \\
1 & a & -1\\
1 & 3 & b\\
\end{...
- 515
1
vote
3
answers
359
views
How can you calculate the rank of an nxn matrix with the given conditions?
Let $A=(a_{i,j})$ a square matrix whose elements are:
$0$ if $i=j$.
$1$ if $j>i$.
$-1$ if $j<i$.
Is there a simple way to find its rank?
- 533
0
votes
1
answer
192
views
Alternating row and column operations for block Gaussian elimination to determine rank.
I am trying to determine the rank of a 6x6 symbolic matrix. The matrix can be represented as follows:
$$
M =
\begin{bmatrix}
A_{ 3 \times 3} & R_{ 3 \times 3}A_{ 3 \times 3}\\
B_{ 3 \times 3} &...
0
votes
1
answer
59
views
Comparing the ranks of leading princial minors of a square symmetric matrix
Let $A$ be a $n$ x $n$ real symmetric matrix and let $A_k$ denotes the $k$-th order leading principal minor matrix of $A$. Prove that for $0 \leq k \leq n-1$:
$$Rank(A_{k+1})\leq Rank(A_k)+2$$
...
- 710
2
votes
1
answer
295
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Finding the rank of matrix that has a parameter
Find the rank of the following matrix. $$A_\lambda = \begin{pmatrix} 2\lambda &-1&2\\ -2&1+\lambda&2-3\lambda\\ -3&-1&5
\end{pmatrix}$$
When finding the rank of matrix, I am ...
- 1,671
0
votes
1
answer
218
views
Determining bases and column ranks using Gauss-Jordan
Consider the following matrix:
\begin{bmatrix} 1 & 0 & -1 & -1 & t-6\\ -t & 0 & 3 & t & 9\\ -1 & 0 & t-6 & 1 & 3\\ 0 & 0 & 0 & t-3 & 0\...
- 4,855
1
vote
2
answers
1k
views
Showing that a Transformation Matrix is injective and surjective.
I have the following linear transformation:
$F: \mathbb P_{3} \to \mathbb R^{3}$ where $\mathbb P_{3}$ is the set of all polynomials with degree at most 3.
$F(p) = (p(0),p(1),p(2))$.
I arrived at ...
- 426
3
votes
1
answer
3k
views
Rank of non-square matrix
Let's be to the point:
How to find rank of a non square matrix like
$$
\begin{pmatrix}
1 & 1 & 1 \\
3 & 2 & 1 \\
1 & 1 & 0 \\
1 & 0 & 0
\end{pmatrix}
$$
I know that ...
- 45
0
votes
0
answers
324
views
Unexpected rank of a matrix
I have a sparse matrix with 100 rows and, when I do a Gauss decomposition, I get a matrix with 90 rows. But if I remove tha last row of the first matrix, now with 99 rows, and I do a Gauss ...
- 103
2
votes
1
answer
809
views
Row & Column Operation to Determine Rank
While evaluating the rank of a matrix is it permissible to apply row and column operations simultaneously on a single matrix? Most of the books that I discussed use either row or column operation (but ...
- 53
0
votes
1
answer
20
views
Method to find the set S of reals $λ$ such as $rg($M-I3)<3 given a matrix
Considering the endomorphism $f$ of $R^3$ of
\begin{bmatrix}
-3 & 5 & -5\\
-4 & 6 & -5\\
-4& 4 &-3
\end{bmatrix}
relatively of the canonical base bc of $R^3$
find the ...
- 1,541
-4
votes
1
answer
156
views
What is the reduced row echelon form of $A$? [closed]
Let $$A = \left( \begin{array}{cccc}
7 & 7 & 9 & -17\\
6 & 6 & 1 & -2 \\
-12 & -12 & -27 & 1 \\
7& 7 & 17 & -15\end{array} \right)$$
What is the ...
- 19