Skip to main content
Mathematics LibreTexts

2.6.1: Exercises 2.6

  1. Last updated
  2. Save as PDF
  • Page ID
    69483

    • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    In Exercises \(\PageIndex{1}\) - \(\PageIndex{8}\), a matrix \(A\) is given. Find \(A^{-1}\) using Theorem 2.6.3, if it exists.

    Exercise \(\PageIndex{1}\)

    \(\left[\begin{array}{cc}{1}&{5}\\{-5}&{-24}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{-24}&{-5}\\{5}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{2}\)

    \(\left[\begin{array}{cc}{1}&{-4}\\{1}&{-3}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{-3}&{4}\\{-1}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{3}\)

    \(\left[\begin{array}{cc}{3}&{0}\\{0}&{7}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{1/3}&{0}\\{0}&{1/7}\end{array}\right]\)

    Exercise \(\PageIndex{4}\)

    \(\left[\begin{array}{cc}{2}&{5}\\{3}&{4}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{-4/7}&{5/7}\\{3/7}&{-2/7}\end{array}\right]\)

    Exercise \(\PageIndex{5}\)

    \(\left[\begin{array}{cc}{1}&{-3}\\{-2}&{6}\end{array}\right]\)

    Answer

    \(A^{-1}\) does not exist.

    Exercise \(\PageIndex{6}\)

    \(\left[\begin{array}{cc}{3}&{7}\\{2}&{4}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{-2}&{7/2}\\{1}&{-3/2}\end{array}\right]\)

    Exercise \(\PageIndex{7}\)

    \(\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{8}\)

    \(\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right]\)

    In Exercises \(\PageIndex{9}\) - \(\PageIndex{28}\), a matrix \(A\) is given. Find \(A^{-1}\) using Key Idea 2.6.1, if it exists.

    Exercise \(\PageIndex{9}\)

    \(\left[\begin{array}{cc}{-2}&{3}\\{1}&{5}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{-5/13}&{3/13}\\{1/13}&{2/13}\end{array}\right]\)

    Exercise \(\PageIndex{10}\)

    \(\left[\begin{array}{cc}{-5}&{-2}\\{9}&{2}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{1/4}&{1/4}\\{-9/8}&{-5/8}\end{array}\right]\)

    Exercise \(\PageIndex{11}\)

    \(\left[\begin{array}{cc}{1}&{2}\\{3}&{4}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{-2}&{1}\\{3/2}&{-1/2}\end{array}\right]\)

    Exercise \(\PageIndex{12}\)

    \(\left[\begin{array}{cc}{5}&{7}\\{5/3}&{7/3}\end{array}\right]\)

    Answer

    \(A^{-1}\) does not exist.

    Exercise \(\PageIndex{13}\)

    \(\left[\begin{array}{ccc}{25}&{-10}&{-4}\\{-18}&{7}&{3}\\{-6}&{2}&{1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{1}&{2}&{-2}\\{0}&{1}&{-3}\\{6}&{10}&{-5}\end{array}\right]\)

    Exercise \(\PageIndex{14}\)

    \(\left[\begin{array}{ccc}{2}&{3}&{4}\\{-3}&{6}&{9}\\{-1}&{9}&{13}\end{array}\right]\)

    Answer

    \(A^{-1}\) does not exist.

    Exercise \(\PageIndex{15}\)

    \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{4}&{1}&{-7}\\{20}&{7}&{-48}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{52}&{-48}&{7}\\{8}&{-7}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{16}\)

    \(\left[\begin{array}{ccc}{-4}&{1}&{5}\\{-5}&{1}&{9}\\{-10}&{2}&{19}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{1}&{-9}&{4}\\{5}&{-26}&{11}\\{0}&{-2}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{17}\)

    \(\left[\begin{array}{ccc}{5}&{-1}&{0}\\{7}&{7}&{1}\\{-2}&{-8}&{-1}\end{array}\right]\)

    Answer

    \(A^{-1}\) does not exist.

    Exercise \(\PageIndex{18}\)

    \(\left[\begin{array}{ccc}{1}&{-5}&{0}\\{-2}&{15}&{4}\\{4}&{-19}&{1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{91}&{5}&{-20}\\{18}&{1}&{-4}\\{-22}&{-1}&{5}\end{array}\right]\)

    Exercise \(\PageIndex{19}\)

    \(\left[\begin{array}{ccc}{25}&{-8}&{0}\\{-78}&{25}&{0}\\{48}&{-15}&{1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{25}&{8}&{0}\\{78}&{25}&{0}\\{-30}&{-9}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{20}\)

    \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{7}&{5}&{8}\\{-2}&{-2}&{-3}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{5}&{-3}&{-8}\\{-4}&{2}&{5}\end{array}\right]\)

    Exercise \(\PageIndex{21}\)

    \(\left[\begin{array}{ccc}{0}&{0}&{1}\\{1}&{0}&{0}\\{0}&{1}&{0}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{0}&{1}&{0}\\{0}&{0}&{1}\\{1}&{0}&{0}\end{array}\right]\)

    Exercise \(\PageIndex{22}\)

    \(\left[\begin{array}{ccc}{0}&{1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{0}&{1}&{0}\\{1}&{0}&{0}\\{0}&{0}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{23}\)

    \(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{-19}&{-9}&{0}&{4}\\{33}&{4}&{1}&{-7}\\{4}&{2}&{0}&{-1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{-3}&{-1}&{0}&{-4}\\{-35}&{-10}&{1}&{-47}\\{-2}&{-2}&{0}&{-9}\end{array}\right]\)

    Exercise \(\PageIndex{24}\)

    \(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{27}&{1}&{0}&{4}\\{18}&{0}&{1}&{4}\\{4}&{0}&{0}&{1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{-11}&{1}&{0}&{-4}\\{-2}&{0}&{1}&{-4}\\{-4}&{0}&{0}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{25}\)

    \(\left[\begin{array}{cccc}{-15}&{45}&{-3}&{4}\\{55}&{-164}&{15}&{-15}\\{-215}&{640}&{-62}&{59}\\{-4}&{12}&{0}&{1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cccc}{28}&{18}&{3}&{-19}\\{5}&{1}&{0}&{-5}\\{4}&{5}&{1}&{0}\\{52}&{60}&{12}&{-15}\end{array}\right]\)

    Exercise \(\PageIndex{26}\)

    \(\left[\begin{array}{cccc}{1}&{0}&{2}&{8}\\{0}&{1}&{0}&{0}\\{0}&{-4}&{-29}&{-110}\\{0}&{-3}&{-5}&{-19}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cccc}{1}&{28}&{-2}&{12}\\{0}&{1}&{0}&{0}\\{0}&{254}&{-19}&{110}\\{0}&{-67}&{5}&{-29}\end{array}\right]\)

    Exercise \(\PageIndex{27}\)

    \(\left[\begin{array}{cccc}{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cccc}{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\end{array}\right]\)

    Exercise \(\PageIndex{28}\)

    \(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{2}&{0}&{0}\\{0}&{0}&{3}&{0}\\{0}&{0}&{0}&{-4}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{1/2}&{0}&{0}\\{0}&{0}&{1/3}&{0}\\{0}&{0}&{0}&{-1/4}\end{array}\right]\)

    In Exercises \(\PageIndex{29}\) - \(\PageIndex{36}\), a matrix \(A\) and a vector \(\vec{b}\) are given. Solve the equation \(A\vec{x}=\vec{b}\) using Theorem 2.6.4.

    Exercise \(\PageIndex{29}\)

    \(A=\left[\begin{array}{cc}{3}&{5}\\{2}&{3}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{21}\\{13}\end{array}\right]\)

    Answer

    \(\vec{x}=\left[\begin{array}{c}{2}\\{3}\end{array}\right]\)

    Exercise \(\PageIndex{30}\)

    \(A=\left[\begin{array}{cc}{1}&{-4}\\{4}&{-15}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{21}\\{77}\end{array}\right]\)

    Answer

    \(\vec{x}=\left[\begin{array}{c}{-7}\\{-7}\end{array}\right]\)

    Exercise \(\PageIndex{31}\)

    \(A=\left[\begin{array}{cc}{9}&{70}\\{-4}&{-31}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-2}\\{1}\end{array}\right]\)

    Answer

    \(\vec{x}=\left[\begin{array}{c}{-8}\\{1}\end{array}\right]\)

    Exercise \(\PageIndex{32}\)

    \(A=\left[\begin{array}{cc}{10}&{-57}\\{3}&{-17}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-14}\\{-4}\end{array}\right]\)

    Answer

    \(\vec{x}=\left[\begin{array}{c}{10}\\{2}\end{array}\right]\)

    Exercise \(\PageIndex{33}\)

    \(A=\left[\begin{array}{ccc}{1}&{2}&{12}\\{0}&{1}&{6}\\{-3}&{0}&{1}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-17}\\{-5}\\{20}\end{array}\right]\)

    Answer

    \(\vec{x}=\left[\begin{array}{c}{-7}\\{1}\\{-1}\end{array}\right]\)

    Exercise \(\PageIndex{34}\)

    \(A=\left[\begin{array}{ccc}{1}&{0}&{-3}\\{8}&{-2}&{-13}\\{12}&{-3}&{-20}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-34}\\{-159}\\{-243}\end{array}\right]\)

    Answer

    \(\vec{x}=\left[\begin{array}{c}{-7}\\{-7}\\{9}\end{array}\right]\)

    Exercise \(\PageIndex{35}\)

    \(A=\left[\begin{array}{ccc}{5}&{0}&{-2}\\{-8}&{1}&{5}\\{-2}&{0}&{1}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{33}\\{-70}\\{-15}\end{array}\right]\)

    Answer

    \(\vec{x}=\left[\begin{array}{c}{3}\\{-1}\\{-9}\end{array}\right]\)

    Exercise \(\PageIndex{36}\)

    \(A=\left[\begin{array}{ccc}{1}&{-6}&{0}\\{0}&{1}&{0}\\{2}&{-8}&{1}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-69}\\{10}\\{-102}\end{array}\right]\)

    Answer

    \(\vec{x}=\left[\begin{array}{c}{-9}\\{10}\\{-4}\end{array}\right]\)

    2.6.1: Exercises 2.6 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    1. Back to top
    • Was this article helpful?

    Recommended articles

    1. Article type
      Section or Page
      License Version
      2.0
    2. Tags
      This page has no tags.