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 1 hour ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
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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 1 hour ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
161
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 1 hour ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
161
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 1 hour ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
161
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 1 hour ago
Martin Sleziak
44.8k9118272
asked 2 hours ago
BaileyBailey
161
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 1 hour ago
Martin Sleziak
44.8k9118272
edited 1 hour ago
Martin Sleziak
44.8k9118272
edited 1 hour ago
Martin Sleziak
44.8k9118272
44.8k9118272
asked 2 hours ago
BaileyBailey
161
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
161
asked 2 hours ago
BaileyBailey
161
161
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.
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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add a comment |
3 Answers
3
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oldest
votes
$begingroup$
You can check directly that
$$
M^{-1}=begin{bmatrix} 0&B^{-1}\ A^{-1}&0end{bmatrix}.
$$
answered 2 hours 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 48 mins ago
FredFred
45.3k1847
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You can check directly that
$$
M^{-1}=begin{bmatrix} 0&B^{-1}\ A^{-1}&0end{bmatrix}.
$$
answered 2 hours 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 2 hours 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 2 hours ago
Martin ArgeramiMartin Argerami
126k1182180
$endgroup$
You can check directly that
$$
M^{-1}=begin{bmatrix} 0&B^{-1}\ A^{-1}&0end{bmatrix}.
$$
answered 2 hours ago
Martin ArgeramiMartin Argerami
126k1182180
answered 2 hours ago
Martin ArgeramiMartin Argerami
126k1182180
answered 2 hours ago
Martin ArgeramiMartin Argerami
126k1182180
answered 2 hours 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 48 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 48 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 48 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 48 mins ago
FredFred
45.3k1847
answered 48 mins ago
FredFred
45.3k1847
answered 48 mins ago
FredFred
45.3k1847
answered 48 mins ago
FredFred
45.3k1847
45.3k1847
add a comment |
add a comment |
Bailey is a new contributor. Be nice, and check out our Code of Conduct.
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