
# 3.3.1: Exercises 3.3


In Exercises $$\PageIndex{1}$$ - $$\PageIndex{8}$$, find the determinant of the $$2\times 2$$ matrix.

Exercise $$\PageIndex{1}$$

$$\left[\begin{array}{cc}{10}&{7}\\{8}&{9}\end{array}\right]$$

$$34$$

Exercise $$\PageIndex{2}$$

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

$$41$$

Exercise $$\PageIndex{3}$$

$$\left[\begin{array}{cc}{-1}&{-7}\\{-5}&{9}\end{array}\right]$$

$$-44$$

Exercise $$\PageIndex{4}$$

$$\left[\begin{array}{cc}{-10}&{-1}\\{-4}&{7}\end{array}\right]$$

$$-74$$

Exercise $$\PageIndex{5}$$

$$\left[\begin{array}{cc}{8}&{10}\\{2}&{-3}\end{array}\right]$$

$$-44$$

Exercise $$\PageIndex{6}$$

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

$$-100$$

Exercise $$\PageIndex{7}$$

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

$$28$$

Exercise $$\PageIndex{8}$$

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

$$11$$

In Exercises $$\PageIndex{9}$$ - $$\PageIndex{12}$$, a matrix $$A$$ is given.

1. Construct the submatrices used to compute the minors $$A_{1,1}$$, $$A_{1,2}$$, and $$A_{1,3}$$.
2. Find the cofactors $$C_{1,1}$$, $$C_{1,2}$$, and $$C_{1,3}$$.

Exercise $$\PageIndex{9}$$

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

1. The submatrices are $$\left[\begin{array}{cc}{7}&{6}\\{6}&{10}\end{array}\right]$$, $$\left[\begin{array}{cc}{3}&{6}\\{1}&{10}\end{array}\right]$$, and $$\left[\begin{array}{cc}{3}&{7}\\{1}&{6}\end{array}\right]$$, respectively.
2. $$C_{1,2} = 34$$, $$C_{1,2} = −24$$, $$C_{1,3} = 11$$

Exercise $$\PageIndex{10}$$

$$\left[\begin{array}{ccc}{-2}&{-9}&{6}\\{-10}&{-6}&{8}\\{0}&{-3}&{-2}\end{array}\right]$$

1. The submatrices are $$\left[\begin{array}{cc}{-6}&{8}\\{-3}&{-2}\end{array}\right]$$, $$\left[\begin{array}{cc}{-10}&{8}\\{0}&{-2}\end{array}\right]$$, and $$\left[\begin{array}{cc}{10}&{-6}\\{0}&{-3}\end{array}\right]$$, respectively.
2. $$C_{1,2} = 36$$, $$C_{1,2} = −20$$, $$C_{1,3} = -30$$

Exercise $$\PageIndex{11}$$

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

1. The submatrices are $$\left[\begin{array}{cc}{3}&{10}\\{3}&{9}\end{array}\right]$$, $$\left[\begin{array}{cc}{-3}&{10}\\{-9}&{9}\end{array}\right]$$, and $$\left[\begin{array}{cc}{-3}&{3}\\{-9}&{3}\end{array}\right]$$, respectively.
2. $$C_{1,2} = -3$$, $$C_{1,2} = −63$$, $$C_{1,3} = 18$$

Exercise $$\PageIndex{12}$$

$$\left[\begin{array}{ccc}{-6}&{-4}&{6}\\{-8}&{0}&{0}\\{-10}&{8}&{-1}\end{array}\right]$$

1. The submatrices are $$\left[\begin{array}{ccc}{-6}&{-4}&{6}\\{-8}&{0}&{0}\\{-10}&{8}&{-1}\end{array}\right]$$, $$\left[\begin{array}{cc}{-8}&{0}\\{-10}&{-1}\end{array}\right]$$, and $$\left[\begin{array}{cc}{-8}&{0}\\{-10}&{8}\end{array}\right]$$, respectively.
2. $$C_{1,2} = 0$$, $$C_{1,2} = −8$$, $$C_{1,3} = -64$$

In Exercises $$\PageIndex{13}$$ – $$\PageIndex{24}$$, find the determinant of the given matrix using cofactor expansion along the first row.

Exercise $$\PageIndex{13}$$

$$\left[\begin{array}{ccc}{3}&{2}&{3}\\{-6}&{1}&{-10}\\{-8}&{-9}&{-9}\end{array}\right]$$

$$-59$$

Exercise $$\PageIndex{14}$$

$$\left[\begin{array}{ccc}{8}&{-9}&{-2}\\{-9}&{9}&{-7}\\{5}&{-1}&{9}\end{array}\right]$$

$$250$$

Exercise $$\PageIndex{15}$$

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

$$15$$

Exercise $$\PageIndex{16}$$

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

$$-52$$

Exercise $$\PageIndex{17}$$

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

$$3$$

Exercise $$\PageIndex{18}$$

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

$$0$$

Exercise $$\PageIndex{19}$$

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

$$0$$

Exercise $$\PageIndex{20}$$

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

$$1$$

Exercise $$\PageIndex{21}$$

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

$$0$$

Exercise $$\PageIndex{22}$$

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

$$2$$

Exercise $$\PageIndex{23}$$

$$\left[\begin{array}{cccc}{-5}&{1}&{0}&{0}\\{-3}&{-5}&{2}&{5}\\{-2}&{4}&{-3}&{4}\\{5}&{4}&{-3}&{3}\end{array}\right]$$

$$-113$$

Exercise $$\PageIndex{24}$$

$$\left[\begin{array}{cccc}{2}&{-1}&{4}&{4}\\{3}&{-3}&{3}&{2}\\{0}&{4}&{-5}&{1}\\{-2}&{-5}&{-2}&{-5}\end{array}\right]$$

$$179$$

Exercise $$\PageIndex{25}$$

Let $$A$$ be a $$2\times 2$$ matrix;

$A=\left[\begin{array}{cc}{a}&{b}\\{c}&{d}\end{array}\right].$

Show why $$\text{det}(A)=ad-bc$$ by computing the cofactor expansion of $$A$$ along the first row.

Hint: $$C_{1,1}=d$$.