Invertible Matrices within a Matrix
$begingroup$
Suppose A, B are invertible matrices of the same size. Show that
$$M = begin{bmatrix} 0& A\ B& 0end{bmatrix}$$ is invertible.
I don't understand how I could show this. I have learned about linear combinations and spanning in my college class, but I don't know how that would help in this case.
linear-algebra matrices inverse block-matrices
edited 54 mins ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
111
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$endgroup$
add a comment |
$begingroup$
Suppose A, B are invertible matrices of the same size. Show that
$$M = begin{bmatrix} 0& A\ B& 0end{bmatrix}$$ is invertible.
I don't understand how I could show this. I have learned about linear combinations and spanning in my college class, but I don't know how that would help in this case.
linear-algebra matrices inverse block-matrices
edited 54 mins ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
111
New contributor
Bailey is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Suppose A, B are invertible matrices of the same size. Show that
$$M = begin{bmatrix} 0& A\ B& 0end{bmatrix}$$ is invertible.
I don't understand how I could show this. I have learned about linear combinations and spanning in my college class, but I don't know how that would help in this case.
linear-algebra matrices inverse block-matrices
edited 54 mins ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
111
New contributor
Bailey is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Suppose A, B are invertible matrices of the same size. Show that
$$M = begin{bmatrix} 0& A\ B& 0end{bmatrix}$$ is invertible.
I don't understand how I could show this. I have learned about linear combinations and spanning in my college class, but I don't know how that would help in this case.
linear-algebra matrices inverse block-matrices
linear-algebra matrices inverse block-matrices
edited 54 mins ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
111
New contributor
Bailey is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 54 mins ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
111
New contributor
Bailey is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 54 mins ago
Martin Sleziak
44.8k9118272
edited 54 mins ago
Martin Sleziak
44.8k9118272
edited 54 mins ago
Martin Sleziak
44.8k9118272
44.8k9118272
asked 2 hours ago
BaileyBailey
111
New contributor
Bailey is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 hours ago
BaileyBailey
111
asked 2 hours ago
BaileyBailey
111
111
New contributor
Bailey is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Bailey is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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3 Answers
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votes
$begingroup$
You can check directly that
$$
M^{-1}=begin{bmatrix} 0&B^{-1}\ A^{-1}&0end{bmatrix}.
$$
answered 1 hour ago
Martin ArgeramiMartin Argerami
126k1182180
$endgroup$
add a comment |
$begingroup$
Note that $M begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} w \ z end{bmatrix}$ iff $A y = w$ and $Bx = z$.
From this you can compute an explicit inverse.
answered 2 hours ago
copper.hatcopper.hat
126k559160
$endgroup$
add a comment |
$begingroup$
Since $A$ and $B$ are invertible, we have
$M begin{bmatrix} x \ y end{bmatrix}=begin{bmatrix} 0 \ 0 end{bmatrix} iff Ay=0$ and $Bx=0 iff x=y=0.$
answered 13 mins ago
FredFred
45.3k1847
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3 Answers
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active
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votes
3 Answers
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active
oldest
votes
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votes
active
oldest
votes
$begingroup$
You can check directly that
$$
M^{-1}=begin{bmatrix} 0&B^{-1}\ A^{-1}&0end{bmatrix}.
$$
answered 1 hour ago
Martin ArgeramiMartin Argerami
126k1182180
$endgroup$
add a comment |
$begingroup$
You can check directly that
$$
M^{-1}=begin{bmatrix} 0&B^{-1}\ A^{-1}&0end{bmatrix}.
$$
answered 1 hour ago
Martin ArgeramiMartin Argerami
126k1182180
$endgroup$
add a comment |
$begingroup$
You can check directly that
$$
M^{-1}=begin{bmatrix} 0&B^{-1}\ A^{-1}&0end{bmatrix}.
$$
answered 1 hour ago
Martin ArgeramiMartin Argerami
126k1182180
$endgroup$
You can check directly that
$$
M^{-1}=begin{bmatrix} 0&B^{-1}\ A^{-1}&0end{bmatrix}.
$$
answered 1 hour ago
Martin ArgeramiMartin Argerami
126k1182180
answered 1 hour ago
Martin ArgeramiMartin Argerami
126k1182180
answered 1 hour ago
Martin ArgeramiMartin Argerami
126k1182180
answered 1 hour ago
Martin ArgeramiMartin Argerami
126k1182180
126k1182180
add a comment |
add a comment |
$begingroup$
Note that $M begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} w \ z end{bmatrix}$ iff $A y = w$ and $Bx = z$.
From this you can compute an explicit inverse.
answered 2 hours ago
copper.hatcopper.hat
126k559160
$endgroup$
add a comment |
$begingroup$
Note that $M begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} w \ z end{bmatrix}$ iff $A y = w$ and $Bx = z$.
From this you can compute an explicit inverse.
answered 2 hours ago
copper.hatcopper.hat
126k559160
$endgroup$
add a comment |
$begingroup$
Note that $M begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} w \ z end{bmatrix}$ iff $A y = w$ and $Bx = z$.
From this you can compute an explicit inverse.
answered 2 hours ago
copper.hatcopper.hat
126k559160
$endgroup$
Note that $M begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} w \ z end{bmatrix}$ iff $A y = w$ and $Bx = z$.
From this you can compute an explicit inverse.
answered 2 hours ago
copper.hatcopper.hat
126k559160
answered 2 hours ago
copper.hatcopper.hat
126k559160
answered 2 hours ago
copper.hatcopper.hat
126k559160
answered 2 hours ago
copper.hatcopper.hat
126k559160
126k559160
add a comment |
add a comment |
$begingroup$
Since $A$ and $B$ are invertible, we have
$M begin{bmatrix} x \ y end{bmatrix}=begin{bmatrix} 0 \ 0 end{bmatrix} iff Ay=0$ and $Bx=0 iff x=y=0.$
answered 13 mins ago
FredFred
45.3k1847
$endgroup$
add a comment |
$begingroup$
Since $A$ and $B$ are invertible, we have
$M begin{bmatrix} x \ y end{bmatrix}=begin{bmatrix} 0 \ 0 end{bmatrix} iff Ay=0$ and $Bx=0 iff x=y=0.$
answered 13 mins ago
FredFred
45.3k1847
$endgroup$
add a comment |
$begingroup$
Since $A$ and $B$ are invertible, we have
$M begin{bmatrix} x \ y end{bmatrix}=begin{bmatrix} 0 \ 0 end{bmatrix} iff Ay=0$ and $Bx=0 iff x=y=0.$
answered 13 mins ago
FredFred
45.3k1847
$endgroup$
Since $A$ and $B$ are invertible, we have
$M begin{bmatrix} x \ y end{bmatrix}=begin{bmatrix} 0 \ 0 end{bmatrix} iff Ay=0$ and $Bx=0 iff x=y=0.$
answered 13 mins ago
FredFred
45.3k1847
answered 13 mins ago
FredFred
45.3k1847
answered 13 mins ago
FredFred
45.3k1847
answered 13 mins ago
FredFred
45.3k1847
45.3k1847
add a comment |
add a comment |
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