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Chapter 1: Q12E (page 39)
Question: There exists a 2x2 matrix such that.
Answer:
True, there exist a2x2 matrix A such that
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Determine which of the matrices in Exercises 7–12areorthogonal. If orthogonal, find the inverse.
11. \(\left( {\begin{aligned}{{}}{2/3}&{2/3}&{1/3}\\0&{1/3}&{ - 2/3}\\{5/3}&{ - 4/3}&{ - 2/3}\end{aligned}} \right)\)
Solve each system in Exercises 1–4 by using elementary row operations on the equations or on the augmented matrix. Follow the systematic elimination procedure.
Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?
In Exercises 33 and 34, Tis a linear transformation from \({\mathbb{R}^2}\) into \({\mathbb{R}^2}\). Show that T is invertible and find a formula for \({T^{ - 1}}\).
34. \(T\left( {{x_1},{x_2}} \right) = \left( {6{x_1} - 8{x_2}, - 5{x_1} + 7{x_2}} \right)\)
Let \(A\) be a \(3 \times 3\) matrix with the property that the linear transformation \({\bf{x}} \mapsto A{\bf{x}}\) maps \({\mathbb{R}^3}\) into \({\mathbb{R}^3}\). Explain why transformation must be one-to-one.
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