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Chapter 2: Q2.7-2Q (page 93)

Use matrix multiplication to find the image of the triangle with data matrix \(D = \left[ {\begin{array}{*{20}{c}}{\bf{5}}&{\bf{2}}&{\bf{4}}\\{\bf{0}}&{\bf{2}}&{\bf{3}}\end{array}} \right]\) under the transformation that reflects points through the y-axis. Sketch both the original triangle and its image.

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}{ - 5}&{ - 2}&{ - 4}\\0&2&3\end{array}} \right]\)

Step by step solution

01

Find the transformed matrix

The matrix of transformation is \(A = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&1\end{array}} \right]\). The transformed data matrix is shown below:

\(\begin{array}{c}AD = \left[ {\begin{array}{*{20}{c}}{ - 1}&0\\0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}5&2&4\\0&2&3\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{ - 5}&{ - 2}&{ - 4}\\0&2&3\end{array}} \right]\end{array}\)

02

Sketch the original triangle and the transformed triangle

The figure below represents the transformed triangle.

So, the transformed matrix is \(\left[ {\begin{array}{*{20}{c}}{ - 5}&{ - 2}&{ - 4}\\0&2&3\end{array}} \right]\).

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Most popular questions from this chapter

2. Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{3}}&{\bf{2}}\\{\bf{7}}&{\bf{4}}\end{aligned}} \right)\).

Exercises 15 and 16 concern arbitrary matrices A, B, and Cfor which the indicated sums and products are defined. Mark each statement True or False. Justify each answer.

15. a. If A and B are \({\bf{2}} \times {\bf{2}}\) with columns \({{\bf{a}}_1},{{\bf{a}}_2}\) and \({{\bf{b}}_1},{{\bf{b}}_2}\) respectively, then \(AB = \left( {\begin{aligned}{*{20}{c}}{{{\bf{a}}_1}{{\bf{b}}_1}}&{{{\bf{a}}_2}{{\bf{b}}_2}}\end{aligned}} \right)\).

b. Each column of ABis a linear combination of the columns of Busing weights from the corresponding column of A.

c. \(AB + AC = A\left( {B + C} \right)\)

d. \({A^T} + {B^T} = {\left( {A + B} \right)^T}\)

e. The transpose of a product of matrices equals the product of their transposes in the same order.

Let \(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&3\\1&4&5\end{aligned}} \right)\), and \(D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&3&0\\0&0&5\end{aligned}} \right)\). Compute \(AD\) and \(DA\). Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a \(3 \times 3\) matrix B, not the identity matrix or the zero matrix, such that \(AB = BA\).

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

36. Write the command(s) that will create a \(6 \times 4\) matrix with random entries. In what range of numbers do the entries lie? Tell how to create a \(3 \times 3\) matrix with random integer entries between \( - {\bf{9}}\) and 9. (Hint:If xis a random number such that 0 < x < 1, then \( - 9.5 < 19\left( {x - .5} \right) < 9.5\).

Prove Theorem 2(b) and 2(c). Use the row-column rule. The \(\left( {i,j} \right)\)- entry in \(A\left( {B + C} \right)\) can be written as \({a_{i1}}\left( {{b_{1j}} + {c_{1j}}} \right) + ... + {a_{in}}\left( {{b_{nj}} + {c_{nj}}} \right)\) or \(\sum\limits_{k = 1}^n {{a_{ik}}\left( {{b_{kj}} + {c_{kj}}} \right)} \).
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