3.5.1: Exercises 3.5

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In Exercises \(\PageIndex{1}\) - \(\PageIndex{12}\), matrices \(A\) and \(\vec{b}\) are given.

Give \(\text{det}(A)\) and \(\text{det}(A_{i})\) for all \(i\).
Use Cramer’s Rule to solve \(A\vec{x}=\vec{b}\). If Cramer’s Rule cannot be used to find the solution, then state whether or not a solution exists.

Exercise \(\PageIndex{1}\)

\(A=\left[\begin{array}{cc}{7}&{-7}\\{-7}&{9}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{28}\\{-26}\end{array}\right]\)

Answer

\(\text{det}(A)=14,\:\text{det}(A_{1})=70,\:\text{det}(A_{2})=14\)
\(\vec{x}=\left[\begin{array}{c}{5}\\{1}\end{array}\right]\)

Exercise \(\PageIndex{2}\)

\(A=\left[\begin{array}{cc}{9}&{5}\\{-4}&{-7}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-45}\\{20}\end{array}\right]\)

Answer

\(\text{det}(A)=-43,\:\text{det}(A_{1})=215,\:\text{det}(A_{2})=0\)
\(\vec{x}=\left[\begin{array}{c}{-5}\\{0}\end{array}\right]\)

Exercise \(\PageIndex{3}\)

\(A=\left[\begin{array}{cc}{-8}&{16}\\{10}&{-20}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-48}\\{60}\end{array}\right]\)

Answer

\(\text{det}(A)=0,\:\text{det}(A_{1})=0,\:\text{det}(A_{2})=0\)
Infinite solutions exist.

Exercise \(\PageIndex{4}\)

\(A=\left[\begin{array}{cc}{0}&{-6}\\{9}&{-10}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{6}\\{-17}\end{array}\right]\)

Answer

\(\text{det}(A)=54,\:\text{det}(A_{1})=-162,\:\text{det}(A_{2})=-54\)
\(\vec{x}=\left[\begin{array}{c}{-3}\\{-1}\end{array}\right]\)

Exercise \(\PageIndex{5}\)

\(A=\left[\begin{array}{cc}{2}&{10}\\{-1}&{3}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{42}\\{19}\end{array}\right]\)

Answer

\(\text{det}(A)=16,\:\text{det}(A_{1})=-64,\:\text{det}(A_{2})=80\)
\(\vec{x}=\left[\begin{array}{c}{-4}\\{5}\end{array}\right]\)

Exercise \(\PageIndex{6}\)

\(A=\left[\begin{array}{cc}{7}&{14}\\{-2}&{-4}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-1}\\{4}\end{array}\right]\)

Answer

\(\text{det}(A)=0,\:\text{det}(A_{1})=-52,\:\text{det}(A_{2})=26\)
No solution exists.

Exercise \(\PageIndex{7}\)

\(A=\left[\begin{array}{ccc}{3}&{0}&{-3}\\{5}&{4}&{4}\\{5}&{5}&{-4}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{24}\\{0}\\{31}\end{array}\right]\)

Answer

\(\text{det}(A)=-123,\:\text{det}(A_{1})=-492,\:\text{det}(A_{2})=123,\:\text{det}(A_{3})=492\)
\(\vec{x}=\left[\begin{array}{c}{4}\\{-1}\\{-4}\end{array}\right]\)

Exercise \(\PageIndex{8}\)

\(A=\left[\begin{array}{ccc}{4}&{9}&{3}\\{-5}&{-2}&{-13}\\{-1}&{10}&{-13}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-28}\\{35}\\{7}\end{array}\right]\)

Answer

\(\text{det}(A)=0,\:\text{det}(A_{1})=0,\:\text{det}(A_{2})=0,\:\text{det}(A_{3})=0\)
Infinite solutions exist.

Exercise \(\PageIndex{9}\)

\(A=\left[\begin{array}{ccc}{4}&{-4}&{0}\\{5}&{1}&{-1}\\{3}&{-1}&{2}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{16}\\{22}\\{8}\end{array}\right]\)

Answer

\(\text{det}(A)=56,\:\text{det}(A_{1})=224,\:\text{det}(A_{2})=0,\:\text{det}(A_{3})=-112\)
\(\vec{x}=\left[\begin{array}{c}{4}\\{0}\\{-2}\end{array}\right]\)

Exercise \(\PageIndex{10}\)

\(A=\left[\begin{array}{ccc}{1}&{0}&{-10}\\{4}&{-3}&{-10}\\{-9}&{6}&{-2}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-40}\\{-94}\\{132}\end{array}\right]\)

Answer

\(\text{det}(A)=96,\:\text{det}(A_{1})=-960,\:\text{det}(A_{2})=768,\:\text{det}(A_{3})=288\)
\(\vec{x}=\left[\begin{array}{c}{-10}\\{8}\\{3}\end{array}\right]\)

Exercise \(\PageIndex{11}\)

\(A=\left[\begin{array}{ccc}{7}&{-4}&{25}\\{-2}&{1}&{-7}\\{9}&{-7}&{34}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{-1}\\{-3}\\{5}\end{array}\right]\)

Answer

\(\text{det}(A)=0,\:\text{det}(A_{1})=147,\:\text{det}(A_{2})=-49,\:\text{det}(A_{3})=-49\)
No solution exists.

Exercise \(\PageIndex{12}\)

\(A=\left[\begin{array}{ccc}{-6}&{-7}&{-7}\\{5}&{4}&{1}\\{5}&{4}&{8}\end{array}\right],\quad\vec{b}=\left[\begin{array}{c}{58}\\{-35}\\{-49}\end{array}\right]\)

Answer

\(\text{det}(A)=77,\:\text{det}(A_{1})=-385,\:\text{det}(A_{2})=-154,\:\text{det}(A_{3})=-154\)
\(\vec{x}=\left[\begin{array}{c}{-5}\\{-2}\\{-2}\end{array}\right]\)