# 3.1.1: Exercises 3.1

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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 $$\PageIndex{1}$$

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

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

##### Exercise $$\PageIndex{2}$$

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

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

##### Exercise $$\PageIndex{3}$$

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

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

##### Exercise $$\PageIndex{4}$$

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

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

##### Exercise $$\PageIndex{5}$$

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

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

##### Exercise $$\PageIndex{6}$$

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

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

##### Exercise $$\PageIndex{7}$$

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

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

##### Exercise $$\PageIndex{8}$$

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

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

##### Exercise $$\PageIndex{9}$$

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

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

##### Exercise $$\PageIndex{10}$$

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

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

##### Exercise $$\PageIndex{11}$$

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

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

##### Exercise $$\PageIndex{12}$$

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

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

##### Exercise $$\PageIndex{13}$$

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

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

##### Exercise $$\PageIndex{14}$$

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

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

##### Exercise $$\PageIndex{15}$$

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

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

##### Exercise $$\PageIndex{16}$$

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

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

##### Exercise $$\PageIndex{17}$$

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

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

##### Exercise $$\PageIndex{18}$$

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

$$A$$ 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 $$\PageIndex{19}$$

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

$$A$$ 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 $$\PageIndex{20}$$

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

$$A$$ 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 $$\PageIndex{21}$$

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

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

##### Exercise $$\PageIndex{22}$$

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

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

##### Exercise $$\PageIndex{23}$$

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

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

##### Exercise $$\PageIndex{24}$$

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

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