2.1.1: Exercises 2.1
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- 63978
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Matrices \(A\) and \(B\) are given below. In Exercises \(\PageIndex{1}\) - \(\PageIndex{6}\), simplify the given expression.
\[A=\left[\begin{array}{cc}{1}&{-1}\\{7}&{4}\end{array}\right]\qquad B=\left[\begin{array}{cc}{-3}&{2}\\{5}&{9}\end{array}\right]\nonumber \]
\(A+B\)
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\(\left[\begin{array}{cc}{-2}&{1}\\{12}&{13}\end{array}\right]\)
\(2A-3B\)
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\(\left[\begin{array}{cc}{11}&{-8}\\{-1}&{-19}\end{array}\right]\)
\(3A-A\)
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\(\left[\begin{array}{cc}{2}&{-2}\\{14}&{8}\end{array}\right]\)
\(4B-2A\)
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\(\left[\begin{array}{cc}{-14}&{10}\\{6}&{28}\end{array}\right]\)
\(3(A-B)+B\)
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\(\left[\begin{array}{cc}{9}&{-7}\\{11}&{-6}\end{array}\right]\)
\(2(A-B)-(A-3B)\)
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\(\left[\begin{array}{cc}{-2}&{1}\\{12}&{13}\end{array}\right]\)
Matrices \(A\) and \(B\) are given below. In Exercises \(\PageIndex{7}\) - \(\PageIndex{10}\), simplify the given expression.
\[A=\left[\begin{array}{c}{3}\\{5}\end{array}\right]\qquad B=\left[\begin{array}{c}{-2}\\{4}\end{array}\right]\nonumber \]
\(4B-2A\)
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\(\left[\begin{array}{c}{-14}\\{6}\end{array}\right]\)
\(-2A+3B\)
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\(\left[\begin{array}{c}{-12}\\{2}\end{array}\right]\)
\(-2A-3A\)
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\(\left[\begin{array}{c}{-15}\\{-25}\end{array}\right]\)
\(-B+3B-2B\)
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\(\left[\begin{array}{c}{0}\\{0}\end{array}\right]\)
Matrices \(A\) and \(B\) are given below. In Exercises \(\PageIndex{11}\) - \(\PageIndex{14}\), find \(X\) that satisfies the equation.
\[A=\left[\begin{array}{cc}{3}&{-1}\\{2}&{5}\end{array}\right]\qquad B=\left[\begin{array}{cc}{1}&{7}\\{3}&{-4}\end{array}\right]\nonumber \]
\(2A+X=B\)
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\(X=\left[\begin{array}{cc}{-5}&{9}\\{-1}&{-14}\end{array}\right]\)
\(A-X=3B\)
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\(X=\left[\begin{array}{cc}{0}&{-22}\\{-7}&{17}\end{array}\right]\)
\(3A+2X=-1B\)
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\(X=\left[\begin{array}{cc}{-5}&{-2}\\{-9/2}&{-11/2}\end{array}\right]\)
\(A-\frac{1}{2}X=-B\)
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\(X=\left[\begin{array}{cc}{8}&{12}\\{10}&{2}\end{array}\right]\)
In Exercises \(\PageIndex{15}\) - \(\PageIndex{21}\), find values for the scalars \(a\) and \(b\) that satisfy the given equation.
\(a\left[\begin{array}{c}{1}\\{2}\end{array}\right]+b\left[\begin{array}{c}{-1}\\{5}\end{array}\right]=\left[\begin{array}{c}{1}\\{9}\end{array}\right]\)
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\(a = 2\), \(b = 1\)
\(a\left[\begin{array}{c}{-3}\\{1}\end{array}\right]+b\left[\begin{array}{c}{8}\\{4}\end{array}\right]=\left[\begin{array}{c}{7}\\{1}\end{array}\right]\)
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\(a = -1\), \(b = 1/2\)
\(a\left[\begin{array}{c}{4}\\{-2}\end{array}\right]+b\left[\begin{array}{c}{-6}\\{3}\end{array}\right]=\left[\begin{array}{c}{10}\\{-5}\end{array}\right]\)
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\(a = 5/2 + 3/2b\)
\(a\left[\begin{array}{c}{1}\\{1}\end{array}\right]+b\left[\begin{array}{c}{-1}\\{3}\end{array}\right]=\left[\begin{array}{c}{5}\\{5}\end{array}\right]\)
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\(a = 5\), \(b = 0\)
\(a\left[\begin{array}{c}{1}\\{3}\end{array}\right]+b\left[\begin{array}{c}{-3}\\{-9}\end{array}\right]=\left[\begin{array}{c}{4}\\{-12}\end{array}\right]\)
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No solution.
\(a\left[\begin{array}{c}{1}\\{2}\\{3}\end{array}\right]+b\left[\begin{array}{c}{1}\\{1}\\{2}\end{array}\right]=\left[\begin{array}{c}{0}\\{-1}\\{-1}\end{array}\right]\)
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\(a=-1\), \(b=1\)
\(a\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right]+b\left[\begin{array}{c}{5}\\{1}\\{2}\end{array}\right]=\left[\begin{array}{c}{3}\\{4}\\{7}\end{array}\right]\)
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No solution.