2.5.1: Exercises 2.5

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In Exercises \(\PageIndex{1}\) - \(\PageIndex{12}\), matrices \(A\) and \(B\) are given. Solve the matrix equation \(AX=B\).

Exercise \(\PageIndex{1}\)

\(A=\left[\begin{array}{cc}{4}&{-1}\\{-7}&{5}\end{array}\right],\)

\(B=\left[\begin{array}{cc}{8}&{-31}\\{-27}&{38}\end{array}\right]\)

Answer: \(X=\left[\begin{array}{cc}{1}&{-9}\\{-4}&{-5}\end{array}\right]\)

Exercise \(\PageIndex{2}\)

\(A=\left[\begin{array}{cc}{1}&{-3}\\{-3}&{6}\end{array}\right],\)

\(B=\left[\begin{array}{cc}{12}&{-10}\\{-27}&{27}\end{array}\right]\)

Answer: \(X=\left[\begin{array}{cc}{3}&{-7}\\{-3}&{1}\end{array}\right]\)

Exercise \(\PageIndex{3}\)

\(A=\left[\begin{array}{cc}{3}&{3}\\{6}&{4}\end{array}\right],\)

\(B=\left[\begin{array}{cc}{15}&{-39}\\{16}&{-66}\end{array}\right]\)

Answer: \(X=\left[\begin{array}{cc}{-2}&{-7}\\{7}&{-6}\end{array}\right]\)

Exercise \(\PageIndex{4}\)

\(A=\left[\begin{array}{cc}{-3}&{-6}\\{4}&{0}\end{array}\right],\)

\(B=\left[\begin{array}{cc}{48}&{-30}\\{0}&{-8}\end{array}\right]\)

Answer: \(X=\left[\begin{array}{cc}{0}&{-2}\\{-8}&{6}\end{array}\right]\)

Exercise \(\PageIndex{5}\)

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

\(B=\left[\begin{array}{ccc}{13}&{4}&{7}\\{22}&{5}&{12}\end{array}\right]\)

Answer: \(X=\left[\begin{array}{ccc}{-5}&{2}&{-3}\\{-4}&{-3}&{-2}\end{array}\right]\)

Exercise \(\PageIndex{6}\)

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

\(B=\left[\begin{array}{ccc}{-2}&{-10}&{19}\\{13}&{2}&{-2}\end{array}\right]\)

Answer: \(X=\left[\begin{array}{ccc}{-1}&{2}&{-4}\\{-6}&{-2}&{3}\end{array}\right]\)

Exercise \(\PageIndex{7}\)

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

Answer: \(X=\left[\begin{array}{cc}{1}&{0}\\{3}&{-1}\end{array}\right]\)

Exercise \(\PageIndex{8}\)

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

Answer: \(X=\left[\begin{array}{cc}{-1/4}&{1/2}\\{3/4}&{-1/2}\end{array}\right]\)

Exercise \(\PageIndex{9}\)

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

\(B=\left[\begin{array}{ccc}{-18}&{2}&{-14}\\{-38}&{18}&{-13}\\{10}&{2}&{-18}\end{array}\right]\)

Answer: \(X=\left[\begin{array}{ccc}{3}&{-3}&{3}\\{2}&{-2}&{-3}\\{-3}&{-1}&{-2}\end{array}\right]\)

Exercise \(\PageIndex{10}\)

\(A=\left[\begin{array}{ccc}{-5}&{-4}&{-1}\\{8}&{-2}&{-3}\\{6}&{1}&{-8}\end{array}\right],\)

\(B=\left[\begin{array}{ccc}{-21}&{-8}&{-19}\\{65}&{-11}&{-10}\\{75}&{-51}&{33}\end{array}\right]\)

Answer: \(X=\left[\begin{array}{ccc}{6}&{1}&{-1}\\{-1}&{-1}&{7}\\{-5}&{7}&{-4}\end{array}\right]\)

Exercise \(\PageIndex{11}\)

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

Answer: \(X=\left[\begin{array}{ccc}{5/3}&{2/3}&{1}\\{-1/3}&{1/6}&{0}\\{1/3}&{1/3}&{0}\end{array}\right]\)

Exercise \(\PageIndex{12}\)

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

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