Q 64E In Exercises 55 through 64, find... [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: Q 64E (page 86)

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 64.
$[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] X = I_{2}$

Short Answer

Expert verified

The matrix that will satisfy the equation will be $[\begin{matrix} \frac{- 5}{3} + e & \frac{4}{3} - 2 e \\ \frac{2}{3} + f & \frac{- 1}{3} - 2 f \\ e & f \end{matrix}]$ , where e,f are constants.

Step by step solution

Assuming the matrix

Since, the product of matrix $2 脳 3$ and matrix X gives the matrix of orderrole="math" localid="1664196259800" $2 脳 2$ then the order of matrix should be $3 脳 2$ .

Let the matrix be $X = [\begin{matrix} a & c \\ b & d \\ e & f \end{matrix}] .$ .

Then the equation will become

$[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] [\begin{matrix} a & c \\ b & d \\ e & f \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Solving the Matrix

For solving the given matrix equation first of multiply the left-hand side matrix.

$[\begin{matrix} a + 2 b + 3 e & c + 2 d + 3 f \\ 4 a + 5 b + 6 e & 4 c + 5 d + 6 f \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Now equating the left hand and right-hand matrix.

$a + 2 b + 3 e = 1 c + 2 d + 3 f = 0 4 a + 5 b + 6 e = 0 4 c + 5 d + 6 f = 1$

On solving the above equations, we will get

$a = \frac{- 5}{3} + e, b = \frac{4}{3} - 2 e, c = \frac{2}{3} + f, d = \frac{- 1}{3} - 2 f, e = e, f = f$

Hence, the matrix X which satisfied the given equation is $[\begin{matrix} \frac{- 5}{3} + e & \frac{4}{3} - 2 e \\ \frac{2}{3} + f & \frac{- 1}{3} - 2 f \\ e & f \end{matrix}]$ , where are constants.

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91影视

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 64.
$[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] X = I_{2}$

Short Answer

Step by step solution

Assuming the matrix

Solving the Matrix

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91影视

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 64.123456X=I2

Short Answer

Step by step solution

Assuming the matrix

Solving the Matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 64.
$[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}] X = I_{2}$