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Mathematics LibreTexts

3.1.1: Exercises 3.1

  • Page ID
    69519
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    In Exercises \(\PageIndex{1}\) - \(\PageIndex{24}\), a matrix \(A\) is given. Find \(A^{T}\); make note if \(A\) is upper/lower triangular, diagonal, symmetric and/or skew symmetric.

    Exercise \(\PageIndex{1}\)

    \(\left[\begin{array}{cc}{-7}&{4}\\{4}&{-6}\end{array}\right]\)

    Answer

    \(A\) is symmetric. \(\left[\begin{array}{cc}{-7}&{4}\\{4}&{-6}\end{array}\right]\)

    Exercise \(\PageIndex{2}\)

    \(\left[\begin{array}{cc}{3}&{1}\\{-7}&{8}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{3}&{-7}\\{1}&{8}\end{array}\right]\)

    Exercise \(\PageIndex{3}\)

    \(\left[\begin{array}{cc}{1}&{0}\\{0}&{9}\end{array}\right]\)

    Answer

    \(A\) is diagonal, as is \(A^{T}\). \(\left[\begin{array}{cc}{1}&{0}\\{0}&{9}\end{array}\right]\)

    Exercise \(\PageIndex{4}\)

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

    Answer

    \(A\) is symmetric. \(\left[\begin{array}{cc}{13}&{-3}\\{-3}&{1}\end{array}\right]\)

    Exercise \(\PageIndex{5}\)

    \(\left[\begin{array}{cc}{-5}&{-9}\\{3}&{1}\\{-10}&{-8}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{-5}&{3}&{-10}\\{-9}&{1}&{-8}\end{array}\right]\)

    Exercise \(\PageIndex{6}\)

    \(\left[\begin{array}{cc}{-2}&{10}\\{1}&{-7}\\{9}&{-2}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{-2}&{1}&{9}\\{10}&{-7}&{-2}\end{array}\right]\)

    Exercise \(\PageIndex{7}\)

    \(\left[\begin{array}{cccc}{4}&{-7}&{-4}&{-9}\\{-9}&{6}&{3}&{-9}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{cc}{4}&{-9}\\{-7}&{6}\\{-4}&{3}\\{-9}&{-9}\end{array}\right]\)

    Exercise \(\PageIndex{8}\)

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

    Answer

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

    Exercise \(\PageIndex{9}\)

    \(\left[\begin{array}{cccc}{-7}&{-8}&{2}&{-3}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{c}{-7}\\{-8}\\{2}\\{-3}\end{array}\right]\)

    Exercise \(\PageIndex{10}\)

    \(\left[\begin{array}{cccc}{-9}&{8}&{2}&{-7}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{c}{-9}\\{8}\\{2}\\{-7}\end{array}\right]\)

    Exercise \(\PageIndex{11}\)

    \(\left[\begin{array}{ccc}{-9}&{4}&{10}\\{6}&{-3}&{-7}\\{-8}&{1}&{-1}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{-9}&{6}&{-8}\\{4}&{-3}&{1}\\{10}&{-7}&{-1}\end{array}\right]\)

    Exercise \(\PageIndex{12}\)

    \(\left[\begin{array}{ccc}{4}&{-5}&{2}\\{1}&{5}&{9}\\{9}&{2}&{3}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{4}&{1}&{9}\\{-5}&{5}&{2}\\{2}&{9}&{3}\end{array}\right]\)

    Exercise \(\PageIndex{13}\)

    \(\left[\begin{array}{ccc}{4}&{0}&{-2}\\{0}&{2}&{3}\\{-2}&{3}&{6}\end{array}\right]\)

    Answer

    \(A\) is symmetric. \(\left[\begin{array}{ccc}{4}&{0}&{-2}\\{0}&{2}&{3}\\{-2}&{3}&{6}\end{array}\right]\)

    Exercise \(\PageIndex{14}\)

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

    Answer

    \(A\) is symmetric. \(\left[\begin{array}{ccc}{0}&{3}&{-2}\\{3}&{-4}&{1}\\{-2}&{1}&{0}\end{array}\right]\)

    Exercise \(\PageIndex{15}\)

    \(\left[\begin{array}{ccc}{2}&{-5}&{-3}\\{5}&{5}&{-6}\\{7}&{-4}&{-10}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{2}&{5}&{7}\\{-5}&{5}&{-4}\\{-3}&{-6}&{-10}\end{array}\right]\)

    Exercise \(\PageIndex{16}\)

    \(\left[\begin{array}{ccc}{0}&{-6}&{1}\\{6}&{0}&{4}\\{-1}&{-4}&{0}\end{array}\right]\)

    Answer

    \(A\) is skew symmetric. \(\left[\begin{array}{ccc}{0}&{-6}&{1}\\{6}&{0}&{4}\\{-1}&{-4}&{0}\end{array}\right]\)

    Exercise \(\PageIndex{17}\)

    \(\left[\begin{array}{ccc}{4}&{2}&{-9}\\{5}&{-4}&{-10}\\{-6}&{6}&{9}\end{array}\right]\)

    Answer

    \(\left[\begin{array}{ccc}{4}&{5}&{-6}\\{2}&{-4}&{6}\\{-9}&{-10}&{9}\end{array}\right]\)

    Exercise \(\PageIndex{18}\)

    \(\left[\begin{array}{ccc}{4}&{0}&{0}\\{-2}&{-7}&{0}\\{4}&{-2}&{5}\end{array}\right]\)

    Answer

    \(A\) is lower triangular and \(A^{T}\) is upper triangular; \(\left[\begin{array}{ccc}{4}&{-2}&{4}\\{0}&{-7}&{-2}\\{0}&{0}&{5}\end{array}\right]\)

    Exercise \(\PageIndex{19}\)

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

    Answer

    \(A\) is upper triangular; \(A^{T}\) is lower triangular. \(\left[\begin{array}{ccc}{-3}&{0}&{0}\\{-4}&{-3}&{0}\\{-5}&{5}&{-3}\end{array}\right]\)

    Exercise \(\PageIndex{20}\)

    \(\left[\begin{array}{cccc}{6}&{-7}&{2}&{6}\\{0}&{-8}&{-1}&{0}\\{0}&{0}&{1}&{-7}\end{array}\right]\)

    Answer

    \(A\) is upper triangular; \(A^{T}\) is lower triangular. \(\left[\begin{array}{ccc}{6}&{0}&{0}\\{-7}&{-8}&{0}\\{2}&{-1}&{1}\\{6}&{0}&{-7}\end{array}\right]\)

    Exercise \(\PageIndex{21}\)

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

    Answer

    \(A\) is diagonal, as is \(A^{T}\). \(\left[\begin{array}{ccc}{1}&{0}&{0}\\{0}&{2}&{0}\\{0}&{0}&{-1}\end{array}\right]\)

    Exercise \(\PageIndex{22}\)

    \(\left[\begin{array}{ccc}{6}&{-4}&{-5}\\{-4}&{0}&{2}\\{-5}&{2}&{-2}\end{array}\right]\)

    Answer

    \(A\) is symmetric. \(\left[\begin{array}{ccc}{6}&{-4}&{-5}\\{-4}&{0}&{2}\\{-5}&{2}&{-2}\end{array}\right]\)

    Exercise \(\PageIndex{23}\)

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

    Answer

    \(A\) is skew symmetric. \(\left[\begin{array}{ccc}{0}&{-1}&{2}\\{1}&{0}&{-4}\\{-2}&{4}&{0}\end{array}\right]\)

    Exercise \(\PageIndex{24}\)

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

    Answer

    \(A\) is upper and lower triangular; it is diagonal; it is both symmetric and skew symmetric. It’s got it all. \(\left[\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{0}\\{0}&{0}&{0}\end{array}\right]\)


    3.1.1: Exercises 3.1 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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