Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Find the determinants of the linear transformations in Exercises 17 through 28.

26. $T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ from the space V of symmetric 2 × 2 matrices to V

Therefore, the determinant of the linear transformations is given by,

$d e t T = d e t B = - 16$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition.

A determinant is a unique number associated with a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Step 2: Given.

Given linear transformation,

$T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$

Step 3: To find determinant.

For $M \in V$ , we have

$M = [\begin{matrix} a & b \\ b & d \end{matrix}]$

So we compute

$T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] [\begin{matrix} a & b \\ b & d \end{matrix}] + [\begin{matrix} a & b \\ b & d \end{matrix}] [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] T (M) = [\begin{matrix} 2 a + 4 b & 2 a + 4 b + 2 d \\ 2 a + 4 b + 2 d & 4 b + 6 d \end{matrix}]$

Since,

$B = \{[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]\}$ is a basis for $V$ , this means that the matrix of $T$ corresponding to $B$ is

$B = [\begin{matrix} 2 & 4 & 0 \\ 2 & 4 & 2 \\ 0 & 4 & 6 \end{matrix}]$

Therefore,

$d e t T = d e t B = - 16$

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Linear Algebra With Applications

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Short Answer

Find the determinants of the linear transformations in Exercises 17 through 28.

26. $T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ from the space V of symmetric 2 × 2 matrices to V

Step by Step Solution

Step 1: Definition.

Step 2: Given.

Step 3: To find determinant.

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Linear Algebra With Applications

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Short Answer

Find the determinants of the linear transformations in Exercises 17 through 28. 26. T(M)=[1223]M+M[1223]from the space V of symmetric 2 × 2 matrices to V

Step by Step Solution

Step 1: Definition.

Step 2: Given.

Step 3: To find determinant.

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Pure Maths

Find the determinants of the linear transformations in Exercises 17 through 28.

26. $T (M) = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] M + M [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$ from the space V of symmetric 2 × 2 matrices to V