
# 3.2.1: Exercises 3.2


In Exercises $$\PageIndex{1}$$ - $$\PageIndex{15}$$, find the trace of the given matrix.

Exercise $$\PageIndex{1}$$

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

$$6$$

Exercise $$\PageIndex{2}$$

$$\left[\begin{array}{cc}{-3}&{-10}\\{-6}&{4}\end{array}\right]$$

$$1$$

Exercise $$\PageIndex{3}$$

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

$$3$$

Exercise $$\PageIndex{4}$$

$$\left[\begin{array}{cc}{-6}&{0}\\{-10}&{9}\end{array}\right]$$

$$3$$

Exercise $$\PageIndex{5}$$

$$\left[\begin{array}{ccc}{-4}&{1}&{1}\\{-2}&{0}&{0}\\{-1}&{-2}&{-5}\end{array}\right]$$

$$-9$$

Exercise $$\PageIndex{6}$$

$$\left[\begin{array}{ccc}{0}&{-3}&{1}\\{5}&{-5}&{5}\\{-4}&{1}&{0}\end{array}\right]$$

$$-5$$

Exercise $$\PageIndex{7}$$

$$\left[\begin{array}{ccc}{-2}&{-3}&{5}\\{5}&{2}&{0}\\{-1}&{-3}&{1}\end{array}\right]$$

$$1$$

Exercise $$\PageIndex{8}$$

$$\left[\begin{array}{ccc}{4}&{2}&{-1}\\{-4}&{1}&{4}\\{0}&{-5}&{5}\end{array}\right]$$

$$10$$

Exercise $$\PageIndex{9}$$

$$\left[\begin{array}{ccc}{2}&{6}&{4}\\{-1}&{8}&{-10}\end{array}\right]$$

Not defined; the matrix must be square.

Exercise $$\PageIndex{10}$$

$$\left[\begin{array}{cc}{6}&{5}\\{2}&{10}\\{3}&{3}\end{array}\right]$$

Not defined; the matrix must be square.

Exercise $$\PageIndex{11}$$

$$\left[\begin{array}{cccc}{-10}&{6}&{-7}&{-9}\\{-2}&{1}&{6}&{-9}\\{0}&{4}&{-4}&{0}\\{-3}&{-9}&{3}&{-10}\end{array}\right]$$

$$-23$$

Exercise $$\PageIndex{12}$$

$$\left[\begin{array}{cccc}{5}&{2}&{2}&{2}\\{-7}&{4}&{-7}&{-3}\\{9}&{-9}&{-7}&{2}\\{-4}&{8}&{-8}&{-2}\end{array}\right]$$

$$0$$

Exercise $$\PageIndex{13}$$

$$I_{4}$$

$$4$$

Exercise $$\PageIndex{14}$$

$$I_{n}$$

$$n$$

Exercise $$\PageIndex{15}$$

A matrix $$A$$ that is skew symmetric.

$$0$$

In Exercises $$\PageIndex{16}$$ - $$\PageIndex{19}$$, verify Theorem 3.2.1 by:

1. Showing that $$\text{tr}(A)+\text{tr}(B)=\text{tr}(A+B)$$ and
2. Showing that $$\text{tr}(AB)=\text{tr}(BA)$$.

Exercise $$\PageIndex{16}$$

$$A=\left[\begin{array}{cc}{1}&{-1}\\{9}&{-6}\end{array}\right],\quad B=\left[\begin{array}{cc}{-1}&{0}\\{-6}&{3}\end{array}\right]$$

1. $$\text{tr}(A)=-5;\:\text{tr}(B)=-4;\:\text{tr}(A+B)=-9$$
2. $$\text{tr}(AB)=23=\text{tr}(BA)$$

Exercise $$\PageIndex{17}$$

$$A=\left[\begin{array}{cc}{0}&{-8}\\{1}&{8}\end{array}\right],\quad B=\left[\begin{array}{cc}{-4}&{5}\\{-4}&{2}\end{array}\right]$$

1. $$\text{tr}(A)=8;\:\text{tr}(B)=-2;\:\text{tr}(A+B)=6$$
2. $$\text{tr}(AB)=53=\text{tr}(BA)$$

Exercise $$\PageIndex{18}$$

$$A=\left[\begin{array}{ccc}{-8}&{-10}&{10}\\{10}&{5}&{-6}\\{-10}&{1}&{3}\end{array}\right],\quad B=\left[\begin{array}{ccc}{-10}&{-4}&{-3}\\{-4}&{-5}&{4}\\{3}&{7}&{3}\end{array}\right]$$

1. $$\text{tr}(A)=0;\:\text{tr}(B)=-12;\:\text{tr}(A+B)=-12$$
2. $$\text{tr}(AB)=86=\text{tr}(BA)$$
Exercise $$\PageIndex{19}$$
$$A=\left[\begin{array}{ccc}{-10}&{7}&{5}\\{7}&{7}&{-5}\\{8}&{-9}&{2}\end{array}\right],\quad B=\left[\begin{array}{ccc}{-3}&{-4}&{9}\\{4}&{-1}&{-9}\\{-7}&{-8}&{10}\end{array}\right]$$
1. $$\text{tr}(A)=-1;\:\text{tr}(B)=6;\:\text{tr}(A+B)=5$$
2. $$\text{tr}(AB)=201=\text{tr}(BA)$$