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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q30E (page 108)

There exists an invertible 2 脳 2 matrix A such that $A^{- 1} = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ .

Short Answer

Expert verified

The statement is false.

Step by step solution

01

Explaining

The claim is that there exists an invertible 2x2 matrix A such that $A^{- 1} = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ . If $A^{- 1} = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ then: $A = {(A^{- 1})}^{- 1}$

$A^{- 1} = {[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]}^{- 1}$

Now we know that $[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ is invertible as long as $d e t [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] 鈮� 0$

02

Conclusion

If we calculate $d e t [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ we have $d e t [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] = 1 - 1 = 0$ .

Thus the inverse does not exist and the statement is false.

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Find the matrices of the transformations Tand Ldefined in Exercise 80.

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