Q55E TRUE OR FALSE?There exists an in... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q55E (page 109)

TRUE OR FALSE?
There exists an invertible matrix S such that $S^{- 1} (\begin{matrix} 0 & - 1 \\ 0 & 0 \end{matrix})$ is a diagonal matrix.

Short Answer

Expert verified

The statement is false.

Step by step solution

Consider the parameters

The inverse of matrix $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ is computed as, role="math" localid="1659352205736" $A^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix})$

Consider the matrix, $S = (\begin{matrix} a & b \\ c & d \end{matrix})$ .

Consider the inverse matrix, $S^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix})$

Find the values of the elements

The product, $S^{- 1} (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) S$ will be,

$S^{- 1} (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) S = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix}) (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) (\begin{matrix} a & b \\ c & d \end{matrix}) = \frac{1}{a d - b c} (\begin{matrix} 0 & d \\ 0 & - c \end{matrix}) (\begin{matrix} a & b \\ c & d \end{matrix}) = \frac{1}{a d - b c} (\begin{matrix} d c & d^{2} \\ - c^{2} & - d c \end{matrix}) = (\begin{matrix} \frac{d c}{a d - b c} & \frac{d^{2}}{a d - b c} \\ \frac{c^{2}}{a d - b c} & \frac{d c}{a d - b c} \end{matrix})$

For the above matrix to be a diagonal matrix, the elements $d, c 鈮� 0$

If this follows, then $\frac{d^{2}}{a d - b c}, \frac{c^{2}}{a d - b c} 鈮� 0$ .

So, $S^{1} (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) S$ cannot be a diagonal matrix.

Hence, the given statement is false.

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91影视

TRUE OR FALSE?
There exists an invertible matrix S such that $S^{- 1} (\begin{matrix} 0 & - 1 \\ 0 & 0 \end{matrix})$ is a diagonal matrix.

Short Answer

Step by step solution

Consider the parameters

Find the values of the elements

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91影视

TRUE OR FALSE?There exists an invertible matrix S such thatS-1(0-100) is a diagonal matrix.

Short Answer

Step by step solution

Consider the parameters

Find the values of the elements

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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TRUE OR FALSE?
There exists an invertible matrix S such that $S^{- 1} (\begin{matrix} 0 & - 1 \\ 0 & 0 \end{matrix})$ is a diagonal matrix.