Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

TRUE OR FALSE? The equation $d e t (A^{T}) = d e t (A)$ holds for all $2 \times 2$ matrices A.

The given statement is true.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of the determinant.

Suppose $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$ is a $2 \times 2$ matrix.

Then the determinant of matrix A is defined as,

$d e t (A) = [\begin{matrix} a & b \\ c & d \end{matrix}] = a d - b c$

Step 2: Check whether the given statement is a true or false.

The given statement is the equation $d e t {(A)}^{T} = d e t (A)$ holds for all $2 \times 2$ matrix A.

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

Then,

$A^{T} = {[\begin{matrix} a & b \\ c & d \end{matrix}]}^{T} A^{T} = [\begin{matrix} a & c \\ b & d \end{matrix}]$

Find $d e t (A^{T}), d e t (A)$

$d e t (A) = |\begin{matrix} a & b \\ c & d \end{matrix}| = a d - b c d e t (A^{T}) = |\begin{matrix} a & c \\ b & d \end{matrix}| c = a d - c d$

This is clear that, the equation $d e t (A^{T}) = d e t (A)$ is holds for all $2 \times 2$ matrix A

Then the given statement is true.

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Linear Algebra With Applications

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Short Answer

TRUE OR FALSE? The equation $d e t (A^{T}) = d e t (A)$ holds for all $2 \times 2$ matrices A.

Step by Step Solution

Step 1: Definition of the determinant.

Step 2: Check whether the given statement is a true or false.

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Linear Algebra With Applications

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Short Answer

TRUE OR FALSE? The equation det(AT)=det(A) holds for all 2×2 matrices A.

Step by Step Solution

Step 1: Definition of the determinant.

Step 2: Check whether the given statement is a true or false.

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TRUE OR FALSE? The equation $d e t (A^{T}) = d e t (A)$ holds for all $2 \times 2$ matrices A.