3.4.1: Exercises 3.4

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

Exercise \(\PageIndex{1}\)

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

Answer: \(84\)

Exercise \(\PageIndex{2}\)

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

Answer: \(48\)

Exercise \(\PageIndex{3}\)

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

Answer: \(0\)

Exercise \(\PageIndex{4}\)

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

Answer: \(60\)

Exercise \(\PageIndex{5}\)

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

Answer: \(10\)

Exercise \(\PageIndex{6}\)

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

Answer: \(-36\)

Exercise \(\PageIndex{7}\)

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

Answer: \(24\)

Exercise \(\PageIndex{8}\)

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

Answer: \(72\)

Exercise \(\PageIndex{9}\)

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

Answer: \(175\)

Exercise \(\PageIndex{10}\)

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

Answer: \(0\)

Exercise \(\PageIndex{11}\)

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

Answer: \(-200\)

Exercise \(\PageIndex{12}\)

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

Answer: \(57\)

Exercise \(\PageIndex{13}\)

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

Answer: \(34\)

Exercise \(\PageIndex{14}\)

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

Answer: \(29\)

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 \(\PageIndex{15}\)

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

\(A=\left[\begin{array}{ccc}{0}&{3}&{5}\\{-2}&{-4}&{-1}\\{3}&{1}&{0}\end{array}\right]\)
\(B=\left[\begin{array}{ccc}{0}&{3}&{5}\\{3}&{1}&{0}\\{8}&{16}&{4}\end{array}\right]\)
\(C=\left[\begin{array}{ccc}{3}&{4}&{5}\\{3}&{1}&{0}\\{-2}&{-4}&{-1}\end{array}\right]\)

Answer

\(\text{det}(A)=41;\:R_{2}\leftrightarrow R_{3}\)
\(\text{det}(B)=164;\:-4R_{3}\to R_{3}\)
\(\text{det}(C)=-41;\:R_{2}+R_{1}\to R_{1}\)

Exercise \(\PageIndex{16}\)

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

\(A=\left[\begin{array}{ccc}{18}&{14}&{16}\\{1}&{3}&{7}\\{6}&{3}&{3}\end{array}\right]\)
\(B=\left[\begin{array}{ccc}{9}&{7}&{8}\\{1}&{3}&{7}\\{96}&{73}&{83}\end{array}\right]\)
\(C=\left[\begin{array}{ccc}{9}&{1}&{6}\\{7}&{3}&{3}\\{8}&{7}&{3}\end{array}\right]\)

Answer

\(\text{det}(A)=90;\:2R_{1}\to R_{1}\)
\(\text{det}(B)=45;\:10R_{1}+R_{3}\to R_{3}\)
\(\text{det}(C)=45;\:C=A^{T}\)

Exercise \(\PageIndex{17}\)

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

\(A=\left[\begin{array}{ccc}{0}&{0}&{4}\\{5}&{1}&{5}\\{4}&{0}&{2}\end{array}\right]\)
\(B=\left[\begin{array}{ccc}{-5}&{-1}&{-5}\\{-4}&{0}&{-2}\\{0}&{0}&{4}\end{array}\right]\)
\(C=\left[\begin{array}{ccc}{15}&{3}&{15}\\{12}&{0}&{6}\\{0}&{0}&{12}\end{array}\right]\)

Answer

\(\text{det}(A)=-16;\:R_{1}\leftrightarrow R_{2}\) then \(R_{1}\leftrightarrow R_{3}\)
\(\text{det}(B)=-16;\: -R_{1}\to R_{1}\) and \(-R_{2}\to R_{2}\)
\(\text{det}(C)=-432;\:C=3 * M\)

Exercise \(\PageIndex{18}\)

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

\(A=\left[\begin{array}{ccc}{1}&{3}&{9}\\{7}&{9}&{3}\\{5}&{4}&{0}\end{array}\right]\)
\(B=\left[\begin{array}{ccc}{5}&{4}&{0}\\{14}&{18}&{6}\\{3}&{9}&{27}\end{array}\right]\)
\(C=\left[\begin{array}{ccc}{-5}&{-4}&{0}\\{-7}&{-9}&{-3}\\{-1}&{-3}&{-9}\end{array}\right]\)

Answer

\(\text{det}(A)=-120;\: R_{1}\leftrightarrow R_{2}\) then \(R_{1}\leftrightarrow R_{3}\) then \(R_{2}\leftrightarrow R_{3}\)
\(\text{det}(B)=720;\:2R_{2}\to R_{2}\) and \(3R_{3}\to R_{3}\)
\(\text{det}(C)=-120;\: C=-M\)

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 \(\PageIndex{19}\)

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

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

Exercise \(\PageIndex{20}\)

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

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

Exercise \(\PageIndex{21}\)

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

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

Exercise \(\PageIndex{22}\)

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

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

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

Exercise \(\PageIndex{23}\)

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

Answer: \(-59\)

Exercise \(\PageIndex{24}\)

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

Answer: \(250\)

Exercise \(\PageIndex{25}\)

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

Answer: \(15\)

Exercise \(\PageIndex{26}\)

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

Answer: \(-52\)

Exercise \(\PageIndex{27}\)

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

Answer: \(3\)

Exercise \(\PageIndex{28}\)

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

Answer: \(0\)

Exercise \(\PageIndex{29}\)

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

Answer: \(0\)

Exercise \(\PageIndex{30}\)

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

Answer: \(1\)

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