(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

(a) Show that for any invertible matrix A,

[itex](A^{-1})^TA^T=I[/itex] and [itex]A^T(A^{-1})^T=I[/itex]

(b) Deduce that A^{T}is invertible and that its inverse is the transpose of [itex]A^{-1}[/itex]

(c) Deduce also that if A is symmetric then A^{-1}is also symmetric.

2. Relevant equations(AB)^{T}=B^{T}A^{T}

3. The attempt at a solution

(a) If A is invertible,

[itex]AA^{-1}=A^{-1}A=I[/itex]

[itex]\Rightarrow (AA^{-1})^T=I^T[/itex]

[itex]\Rightarrow (AA^{-1})^T=I[/itex]

[itex]\Rightarrow (A^{-1})^TA^T=I[/itex]

Now for part (b) and (c)

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# Homework Help: Does this work. Matrices

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