2.1.1: Exercises 2.1

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Matrices \(A\) and \(B\) are given below. In Exercises \(\PageIndex{1}\) - \(\PageIndex{6}\), simplify the given expression.

\[A=\left[\begin{array}{cc}{1}&{-1}\\{7}&{4}\end{array}\right]\qquad B=\left[\begin{array}{cc}{-3}&{2}\\{5}&{9}\end{array}\right]\nonumber \]

Exercise \(\PageIndex{1}\)

\(A+B\)

Answer: \(\left[\begin{array}{cc}{-2}&{1}\\{12}&{13}\end{array}\right]\)

Exercise \(\PageIndex{2}\)

\(2A-3B\)

Answer: \(\left[\begin{array}{cc}{11}&{-8}\\{-1}&{-19}\end{array}\right]\)

Exercise \(\PageIndex{3}\)

\(3A-A\)

Answer: \(\left[\begin{array}{cc}{2}&{-2}\\{14}&{8}\end{array}\right]\)

Exercise \(\PageIndex{4}\)

\(4B-2A\)

Answer: \(\left[\begin{array}{cc}{-14}&{10}\\{6}&{28}\end{array}\right]\)

Exercise \(\PageIndex{5}\)

\(3(A-B)+B\)

Answer: \(\left[\begin{array}{cc}{9}&{-7}\\{11}&{-6}\end{array}\right]\)

Exercise \(\PageIndex{6}\)

\(2(A-B)-(A-3B)\)

Answer: \(\left[\begin{array}{cc}{-2}&{1}\\{12}&{13}\end{array}\right]\)

Matrices \(A\) and \(B\) are given below. In Exercises \(\PageIndex{7}\) - \(\PageIndex{10}\), simplify the given expression.

\[A=\left[\begin{array}{c}{3}\\{5}\end{array}\right]\qquad B=\left[\begin{array}{c}{-2}\\{4}\end{array}\right]\nonumber \]

Exercise \(\PageIndex{7}\)

\(4B-2A\)

Answer: \(\left[\begin{array}{c}{-14}\\{6}\end{array}\right]\)

Exercise \(\PageIndex{8}\)

\(-2A+3B\)

Answer: \(\left[\begin{array}{c}{-12}\\{2}\end{array}\right]\)

Exercise \(\PageIndex{9}\)

\(-2A-3A\)

Answer: \(\left[\begin{array}{c}{-15}\\{-25}\end{array}\right]\)

Exercise \(\PageIndex{10}\)

\(-B+3B-2B\)

Answer: \(\left[\begin{array}{c}{0}\\{0}\end{array}\right]\)

Matrices \(A\) and \(B\) are given below. In Exercises \(\PageIndex{11}\) - \(\PageIndex{14}\), find \(X\) that satisfies the equation.

\[A=\left[\begin{array}{cc}{3}&{-1}\\{2}&{5}\end{array}\right]\qquad B=\left[\begin{array}{cc}{1}&{7}\\{3}&{-4}\end{array}\right]\nonumber \]

Exercise \(\PageIndex{11}\)

\(2A+X=B\)

Answer: \(X=\left[\begin{array}{cc}{-5}&{9}\\{-1}&{-14}\end{array}\right]\)

Exercise \(\PageIndex{12}\)

\(A-X=3B\)

Answer: \(X=\left[\begin{array}{cc}{0}&{-22}\\{-7}&{17}\end{array}\right]\)

Exercise \(\PageIndex{13}\)

\(3A+2X=-1B\)

Answer: \(X=\left[\begin{array}{cc}{-5}&{-2}\\{-9/2}&{-11/2}\end{array}\right]\)

Exercise \(\PageIndex{14}\)

\(A-\frac{1}{2}X=-B\)

Answer: \(X=\left[\begin{array}{cc}{8}&{12}\\{10}&{2}\end{array}\right]\)

In Exercises \(\PageIndex{15}\) - \(\PageIndex{21}\), find values for the scalars \(a\) and \(b\) that satisfy the given equation.

Exercise \(\PageIndex{15}\)

\(a\left[\begin{array}{c}{1}\\{2}\end{array}\right]+b\left[\begin{array}{c}{-1}\\{5}\end{array}\right]=\left[\begin{array}{c}{1}\\{9}\end{array}\right]\)

Answer: \(a = 2\), \(b = 1\)

Exercise \(\PageIndex{16}\)

\(a\left[\begin{array}{c}{-3}\\{1}\end{array}\right]+b\left[\begin{array}{c}{8}\\{4}\end{array}\right]=\left[\begin{array}{c}{7}\\{1}\end{array}\right]\)

Answer: \(a = -1\), \(b = 1/2\)

Exercise \(\PageIndex{17}\)

\(a\left[\begin{array}{c}{4}\\{-2}\end{array}\right]+b\left[\begin{array}{c}{-6}\\{3}\end{array}\right]=\left[\begin{array}{c}{10}\\{-5}\end{array}\right]\)

Answer: \(a = 5/2 + 3/2b\)

Exercise \(\PageIndex{18}\)

\(a\left[\begin{array}{c}{1}\\{1}\end{array}\right]+b\left[\begin{array}{c}{-1}\\{3}\end{array}\right]=\left[\begin{array}{c}{5}\\{5}\end{array}\right]\)

Answer: \(a = 5\), \(b = 0\)

Exercise \(\PageIndex{19}\)

\(a\left[\begin{array}{c}{1}\\{3}\end{array}\right]+b\left[\begin{array}{c}{-3}\\{-9}\end{array}\right]=\left[\begin{array}{c}{4}\\{-12}\end{array}\right]\)

Answer: No solution.

Exercise \(\PageIndex{20}\)

\(a\left[\begin{array}{c}{1}\\{2}\\{3}\end{array}\right]+b\left[\begin{array}{c}{1}\\{1}\\{2}\end{array}\right]=\left[\begin{array}{c}{0}\\{-1}\\{-1}\end{array}\right]\)

Answer: \(a=-1\), \(b=1\)

Exercise \(\PageIndex{21}\)

\(a\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right]+b\left[\begin{array}{c}{5}\\{1}\\{2}\end{array}\right]=\left[\begin{array}{c}{3}\\{4}\\{7}\end{array}\right]\)

Answer: No solution.