3.4.1: Exercises 3.4


In Exercises $$\PageIndex{1}$$ – $$\PageIndex{14}$$, find the determinant of the given matrix using cofactor expansion along any row or column you choose.

Exercise $$\PageIndex{1}$$

$$\left[\begin{array}{ccc}{1}&{2}&{3}\\{-5}&{0}&{3}\\{4}&{0}&{6}\end{array}\right]$$

$$84$$

Exercise $$\PageIndex{2}$$

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

$$48$$

Exercise $$\PageIndex{3}$$

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

$$0$$

Exercise $$\PageIndex{4}$$

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

$$60$$

Exercise $$\PageIndex{5}$$

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

$$10$$

Exercise $$\PageIndex{6}$$

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

$$-36$$

Exercise $$\PageIndex{7}$$

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

$$24$$

Exercise $$\PageIndex{8}$$

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

$$72$$

Exercise $$\PageIndex{9}$$

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

$$175$$

Exercise $$\PageIndex{10}$$

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

$$0$$

Exercise $$\PageIndex{11}$$

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

$$-200$$

Exercise $$\PageIndex{12}$$

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

$$57$$

Exercise $$\PageIndex{13}$$

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

$$34$$

Exercise $$\PageIndex{14}$$

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

$$29$$

In Exercises $$\PageIndex{15}$$ - $$\PageIndex{18}$$, a matrix $$M$$ and $$\text{det}(M)$$ are given. Matrices $$A$$, $$B$$ and $$C$$ are formed by performing operations on $$M$$. Determine the determinants of $$A$$, $$B$$ and $$C$$ using Theorems 3.4.2 and 3.4.3, and indicate the operations used to form $$A$$, $$B$$ and $$C$$.

Exercise $$\PageIndex{15}$$

$$M=\left[\begin{array}{ccc}{0}&{3}&{5}\\{3}&{1}&{0}\\{-2}&{-4}&{-1}\end{array}\right],\quad \text{det}(M)=-41.$$

1. $$A=\left[\begin{array}{ccc}{0}&{3}&{5}\\{-2}&{-4}&{-1}\\{3}&{1}&{0}\end{array}\right]$$
2. $$B=\left[\begin{array}{ccc}{0}&{3}&{5}\\{3}&{1}&{0}\\{8}&{16}&{4}\end{array}\right]$$
3. $$C=\left[\begin{array}{ccc}{3}&{4}&{5}\\{3}&{1}&{0}\\{-2}&{-4}&{-1}\end{array}\right]$$
1. $$\text{det}(A)=41;\:R_{2}\leftrightarrow R_{3}$$
2. $$\text{det}(B)=164;\:-4R_{3}\to R_{3}$$
3. $$\text{det}(C)=-41;\:R_{2}+R_{1}\to R_{1}$$
Exercise $$\PageIndex{16}$$

$$M=\left[\begin{array}{ccc}{9}&{7}&{8}\\{1}&{3}&{7}\\{6}&{3}&{3}\end{array}\right],\quad \text{det}(M)=45.$$

1. $$A=\left[\begin{array}{ccc}{18}&{14}&{16}\\{1}&{3}&{7}\\{6}&{3}&{3}\end{array}\right]$$
2. $$B=\left[\begin{array}{ccc}{9}&{7}&{8}\\{1}&{3}&{7}\\{96}&{73}&{83}\end{array}\right]$$
3. $$C=\left[\begin{array}{ccc}{9}&{1}&{6}\\{7}&{3}&{3}\\{8}&{7}&{3}\end{array}\right]$$
1. $$\text{det}(A)=90;\:2R_{1}\to R_{1}$$
2. $$\text{det}(B)=45;\:10R_{1}+R_{3}\to R_{3}$$
3. $$\text{det}(C)=45;\:C=A^{T}$$
Exercise $$\PageIndex{17}$$

$$M=\left[\begin{array}{ccc}{5}&{1}&{5}\\{4}&{0}&{2}\\{0}&{0}&{4}\end{array}\right],\quad \text{det}(M)=-16.$$

1. $$A=\left[\begin{array}{ccc}{0}&{0}&{4}\\{5}&{1}&{5}\\{4}&{0}&{2}\end{array}\right]$$
2. $$B=\left[\begin{array}{ccc}{-5}&{-1}&{-5}\\{-4}&{0}&{-2}\\{0}&{0}&{4}\end{array}\right]$$
3. $$C=\left[\begin{array}{ccc}{15}&{3}&{15}\\{12}&{0}&{6}\\{0}&{0}&{12}\end{array}\right]$$
1. $$\text{det}(A)=-16;\:R_{1}\leftrightarrow R_{2}$$ then $$R_{1}\leftrightarrow R_{3}$$
2. $$\text{det}(B)=-16;\: -R_{1}\to R_{1}$$ and $$-R_{2}\to R_{2}$$
3. $$\text{det}(C)=-432;\:C=3 * M$$
Exercise $$\PageIndex{18}$$

$$M=\left[\begin{array}{ccc}{5}&{4}&{0}\\{7}&{9}&{3}\\{1}&{3}&{9}\end{array}\right],\quad \text{det}(M)=120.$$

1. $$A=\left[\begin{array}{ccc}{1}&{3}&{9}\\{7}&{9}&{3}\\{5}&{4}&{0}\end{array}\right]$$
2. $$B=\left[\begin{array}{ccc}{5}&{4}&{0}\\{14}&{18}&{6}\\{3}&{9}&{27}\end{array}\right]$$
3. $$C=\left[\begin{array}{ccc}{-5}&{-4}&{0}\\{-7}&{-9}&{-3}\\{-1}&{-3}&{-9}\end{array}\right]$$
1. $$\text{det}(A)=-120;\: R_{1}\leftrightarrow R_{2}$$ then $$R_{1}\leftrightarrow R_{3}$$ then $$R_{2}\leftrightarrow R_{3}$$
2. $$\text{det}(B)=720;\:2R_{2}\to R_{2}$$ and $$3R_{3}\to R_{3}$$
3. $$\text{det}(C)=-120;\: C=-M$$

In Exercises $$\PageIndex{19}$$ - $$\PageIndex{22}$$, matrices $$A$$ and $$B$$ are given. Verify part 3 of Theorem 3.4.3 by computing $$\text{det}(A)$$, $$\text{det}(B)$$ and $$\text{det}(AB)$$.

Exercise $$\PageIndex{19}$$

$$A=\left[\begin{array}{cc}{2}&{0}\\{1}&{2}\end{array}\right],\quad B=\left[\begin{array}{cc}{0}&{-4}\\{1}&{3}\end{array}\right]$$

$$\text{det}(A)=4,\:\text{det}(B)=4,\:\text{det}(AB)=16$$

Exercise $$\PageIndex{20}$$

$$A=\left[\begin{array}{cc}{3}&{-1}\\{4}&{1}\end{array}\right],\quad B=\left[\begin{array}{cc}{-4}&{-1}\\{-5}&{3}\end{array}\right]$$

$$\text{det}(A)=7,\:\text{det}(B)=-17,\:\text{det}(AB)=-119$$

Exercise $$\PageIndex{21}$$

$$A=\left[\begin{array}{cc}{-4}&{4}\\{5}&{-2}\end{array}\right],\quad B=\left[\begin{array}{cc}{-3}&{-4}\\{5}&{-3}\end{array}\right]$$

$$\text{det}(A)=-12,\:\text{det}(B)=29,\:\text{det}(AB)=-348$$

Exercise $$\PageIndex{22}$$

$$A=\left[\begin{array}{cc}{-3}&{-1}\\{2}&{-3}\end{array}\right],\quad B=\left[\begin{array}{cc}{0}&{0}\\{4}&{-4}\end{array}\right]$$

$$\text{det}(A)=11,\:\text{det}(B)=0,\:\text{det}(AB)=0$$

In Exercises $$\PageIndex{23}$$ - $$\PageIndex{30}$$, find the determinant of the given matrix using Key Idea 3.4.2.

Exercise $$\PageIndex{23}$$

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

$$-59$$

Exercise $$\PageIndex{24}$$

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

$$250$$

Exercise $$\PageIndex{25}$$

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

$$15$$

Exercise $$\PageIndex{26}$$

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

$$-52$$

Exercise $$\PageIndex{27}$$

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

$$3$$

Exercise $$\PageIndex{28}$$

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

$$0$$

Exercise $$\PageIndex{29}$$

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

$$0$$

Exercise $$\PageIndex{30}$$

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

$$1$$