Q16E TRUE OR FALSE?There exists a mat... [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: Q16E (page 108)

TRUE OR FALSE?
There exists a matrix A such that $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] A [\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ .

Short Answer

Expert verified

The statement is true.

Step by step solution

Compute the matrix

The inverse of $A = (\begin{matrix} a & c \\ b & d \end{matrix})$ matrix is $A^{- 1} = \frac{1}{a d - b c} (\begin{matrix} d & - b \\ - c & a \end{matrix})$ , .

Consider the matrix.

$(\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}) A (\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}) = (\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})$

Let $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ .

Compute the matrix

Now, find the product as follows:

$(\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}) A (\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}) = (\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}) 鈬� (\begin{matrix} (a + 2 c) . 5 + (b + 2 d) . 7 & (a + 2 c) . 6 + (b + 2 d) . 8 \\ (3 a + 4 c) . 5 + (3 b + 4 d) . 7 & (3 a + 4 c) . 6 (3 b + 4 d) . 8 \end{matrix}) = (\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}) 鈭� a = \frac{1}{2}, b = - \frac{1}{2}, c = - \frac{1}{2}, d = \frac{1}{2}$

Therefore, the matrix is,

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

Hence, the given statement is true.

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91影视

TRUE OR FALSE?
There exists a matrix A such that $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] A [\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ .

Short Answer

Step by step solution

Compute the matrix

Compute the matrix

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91影视

TRUE OR FALSE?There exists a matrix A such that[1234]A[5678]=[1111] .

Short Answer

Step by step solution

Compute the matrix

Compute the matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Theoretical and Mathematical Physics

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Applied Mathematics

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Study anywhere. Anytime. Across all devices.

TRUE OR FALSE?
There exists a matrix A such that $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] A [\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}] = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ .