Solved Problems for Chapter 4

Using only the properties given in Theorem th:propertiesofaddition and Theorem th:propertiesscalarmult, show that the additive inverse of , , is unique.

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Suppose is also an additive inverse of . Then

Using only the properties given in Theorem th:propertiesofaddition and Theorem th:propertiesscalarmult, show that the zero matrix, , is unique.

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Suppose is also an additive identity. Then

Using only the properties given in Theorem th:propertiesofaddition and Theorem th:propertiesscalarmult, show that

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Now add to both sides. Then .

Using only the properties given in Theorem th:propertiesofaddition and Theorem th:propertiesscalarmult, as well as previous problems, show

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Therefore, from the uniqueness of the additive inverse proved in the above Problem addinvrstunique, it follows that .

Consider the matrices .

Find the following if possible. If it is not possible explain why.

(a)

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(b)

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(c)

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Not possible

(d)

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(e)

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Not possible

(f)

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Not possible

Let . Find all matrices, such that

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Solution is: so the matrices are of the form

Let Is it possible to choose such that If so, what should equal?

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Thus you must have . Therefore .

Find matrices and such that and but .

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Let

Suppose and is an invertible matrix. Does it follow that Explain why or why not.

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Yes . Multiply on the left by .

Let \begin{equation*} A=\left [ \begin{array}{rrr} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 1 & 0 & 2 \end{array} \right ] \end{equation*} Find if possible. If does not exist, explain why.

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Let \begin{equation*} A=\left [ \begin{array}{rrr} 1 & 2 & 3 \\ 2 & 1 & 4 \\ 4 & 5 & 10 \end{array} \right ] \end{equation*} Find if possible. If does not exist, explain why.

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The reduced row echelon form is . There is no inverse.

Show that if exists for an matrix, then it is unique. That is, if and then

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Show that if is an invertible matrix, then so is and

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You need to show that acts like the inverse of because from uniqueness in the above problem, this will imply it is the inverse. From properties of the transpose, Hence and this last matrix exists.

Show by verifying that \begin{equation*} AB\left ( B^{-1}A^{-1}\right ) =I \end{equation*} and \begin{equation*} B^{-1}A^{-1}\left ( AB\right ) =I \end{equation*}

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Find an factorization of the coefficient matrix and use it to solve the system of equations. \begin{equation*} \begin{array}{c} x+2y=5 \\ 2x+3y=6 \end{array} \end{equation*}

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An factorization of the coefficient matrix is First solve which gives Then solve which says that and

Find an factorization of the coefficient matrix and use it to solve the system of equations. \begin{equation*} \begin{array}{c} x+2y+z=1 \\ y+3z=2 \\ 2x+3y=6 \end{array} \end{equation*}

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An factorization of the coefficient matrix is First solve which yields . Next solve This yields

Bibliography

These problems come from the end of Chapter 2 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, pp. 90–98, 104–106.

2024-09-06 23:26:18