Show that the determinant of the matrix is equal to the determinant of its transpose by expanding along the third row of $ A $ and the third column of $ A^T $.\[A = \begin{bmatrix} 2 & -2 & -3 \\ 1 & 0 & 5 \\ 5 & -1 & 0 \end{bmatrix}\]---Write an expression for $|A|$ by expanding along the third row of $ A $.\[|A| = \begin{vmatrix} \Box & \Box & \Box \\ \Box & \Box & \Box \end{vmatrix} + \begin{vmatrix} \Box & \Box \\ \Box & \Box \end{vmatrix}\](Type the terms of your expression in the same order as they appear in the original expression.)

Answered: Show that the determinant of the matrix…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Show that the determinant of the matrix is equal to the determinant of its transpose by expanding along the third row of $ A $ and the third column of $ A^T $.

\[
A = \begin{bmatrix} 2 & -2 & -3 \\ 1 & 0 & 5 \\ 5 & -1 & 0 \end{bmatrix}
\]

---

Write an expression for $|A|$ by expanding along the third row of $ A $.

\[
|A| = \begin{vmatrix} \Box & \Box & \Box \\ \Box & \Box & \Box \end{vmatrix} + \begin{vmatrix} \Box & \Box \\ \Box & \Box \end{vmatrix}
\]

(Type the terms of your expression in the same order as they appear in the original expression.)

Expert Solution

Step 1: Solution

Given matrix:

$A equals open square brackets table row 2 cell negative 2 end cell cell negative 3 end cell row 1 0 5 row 5 cell negative 1 end cell 0 end table close square brackets$

We have to show that the determinant of the matrix is equal to the determinant of its transpose by expanding along the third row of $A$ and the third column of $A to the power of T$

Notation: $D e t left parenthesis A right parenthesis equals open vertical bar A close vertical bar$