- #1
mbrmbrg
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[SOLVED] Linear Algebra: given adj(A) find A
If [tex]adj\mathbb A = \left(\begin{array}{ccc}1&0&1\\1&-1&0\\0&2&1\end{array}\right)[/tex], find A. Briefly justify your algorithm.
[tex]adj\mathbb A=(det\mathbb A)(\mathbb A^{-1})[/tex]
[tex]adj\mathbb A=(det\mathbb A)(\mathbb A^{-1})[/tex]
invert both sides to get:
[tex](adj\mathbb A)^{-1} = [(det\mathbb A)(\mathbb A^{-1})]^{-1}[/tex]
[tex](adj\mathbb A)^{-1} = (\mathbb A^{-1})^{-1}(det\mathbb A)^{-1}[/tex]
[tex](adj\mathbb A)^{-1} = (\mathbb A)(\frac{1}{det\mathbb A})[/tex]
[tex]\mathbb A = (det\mathbb A)(adj\mathbb A)^{-1}[/tex]
My, isn't that nice.
I computed [tex](adj\mathbb A)^{-1}[/tex], and found it to be
[tex]\left(\begin{array}{ccc}-1&2&1\\-1&1&1\\2&-2&-1\end{array}\right)[/tex]
But I have no clue how to find detA, so I'm stuck with one equation:
[tex]\mathbb A = (det\mathbb A)(adj\mathbb A)^{-1}[/tex]
and two unknowns:
[tex]\mathbb A[/tex] and [tex]det\mathbb A[/tex]
Homework Statement
If [tex]adj\mathbb A = \left(\begin{array}{ccc}1&0&1\\1&-1&0\\0&2&1\end{array}\right)[/tex], find A. Briefly justify your algorithm.
Homework Equations
[tex]adj\mathbb A=(det\mathbb A)(\mathbb A^{-1})[/tex]
The Attempt at a Solution
[tex]adj\mathbb A=(det\mathbb A)(\mathbb A^{-1})[/tex]
invert both sides to get:
[tex](adj\mathbb A)^{-1} = [(det\mathbb A)(\mathbb A^{-1})]^{-1}[/tex]
[tex](adj\mathbb A)^{-1} = (\mathbb A^{-1})^{-1}(det\mathbb A)^{-1}[/tex]
[tex](adj\mathbb A)^{-1} = (\mathbb A)(\frac{1}{det\mathbb A})[/tex]
[tex]\mathbb A = (det\mathbb A)(adj\mathbb A)^{-1}[/tex]
My, isn't that nice.
I computed [tex](adj\mathbb A)^{-1}[/tex], and found it to be
[tex]\left(\begin{array}{ccc}-1&2&1\\-1&1&1\\2&-2&-1\end{array}\right)[/tex]
But I have no clue how to find detA, so I'm stuck with one equation:
[tex]\mathbb A = (det\mathbb A)(adj\mathbb A)^{-1}[/tex]
and two unknowns:
[tex]\mathbb A[/tex] and [tex]det\mathbb A[/tex]
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