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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

In exercise 11 and 12, the matrices are all \(n \times n\). Each part of the exercise is an implication of the form 鈥淚f 鈥渟tatement 1鈥� then 鈥渟tatement 2鈥�.鈥滿ark the implication as True if the truth of 鈥渟tatement 2鈥漚lways follows whenever 鈥渟tatement 1鈥� happens to be true. An implication is False if there is an instance in which 鈥渟tatement 2鈥� is false but 鈥渟tatement 1鈥� is true. Justify each answer.

a. If the equation \[A{\bf{x}} = {\bf{0}}\] has only the trivial solution, then \(A\) is row equivalent to the \(n \times n\) identity matrix.

b. If the columns of \(A\) span \({\mathbb{R}^n}\), then the columns are linearly independent.

c. If \(A\) is an \(n \times n\) matrix, then the equation \(A{\bf{x}} = {\bf{b}}\) has at least one solution for each \({\bf{b}}\) in \({\mathbb{R}^n}\).

d. If the equation \[A{\bf{x}} = {\bf{0}}\] has a non trivial solution, then \[A\] has fewer than \(n\) pivot positions.

e. If \({A^T}\) is not invertible, then \(A\) is not invertible.

Suppose the first two columns, \({{\bf{b}}_1}\) and \({{\bf{b}}_2}\), of Bare equal. What can you say about the columns of AB(if ABis defined)? Why?

Suppose \(CA = {I_n}\)(the \(n \times n\) identity matrix). Show that the equation \(Ax = 0\) has only the trivial solution. Explain why Acannot have more columns than rows.

Suppose Aand Bare \(n \times n\), Bis invertible, and ABis invertible. Show that Ais invertible. (Hint: Let C=AB, and solve this equation for A.)

In Exercise 10 mark each statement True or False. Justify each answer.

10. a. A product of invertible \(n \times n\) matrices is invertible, and the inverse of the product of their inverses in the same order.

b. If A is invertible, then the inverse of \({A^{ - {\bf{1}}}}\) is A itself.

c. If \(A = \left( {\begin{aligned}{*{20}{c}}a&b\\c&d\end{aligned}} \right)\) and \(ad = bc\), then A is not invertible.

d. If A can be row reduced to the identity matrix, then A must be invertible.

e. If A is invertible, then elementary row operations that reduce A to the identity \({I_n}\) also reduce \({A^{ - {\bf{1}}}}\) to \({I_n}\).
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