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Exercise 1

Avg difficulty 2.5

Consider the matrix A=[3−1140231−210−23012]A = \left[\begin{array}{rrrr} 3 & -1 & 1 & 4 \\ 0 & 2 & 3 & 1 \\ -2 & 1 & 0 & -2 \\ 3 & 0 & 1 & 2 \\ \end{array}\right].

  1. Explain why AA has an inverse.

  2. Find the inverse of AA by augmenting by the identity II to form [AI]\left[\begin{array}{r|r}A & I \end{array}\right].

  3. Use your inverse to solve the equation Ax=[32−3−1]A\xvec = \fourvec{3}{2}{-3}{-1}.

Austin, Understanding Linear Algebra, §3.1 Invertibility, Exercise 1

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.