Determinant Equality Explained without Evaluation | Boas 3rd Ed. HW Question 7

[iamthegelo]

Homework Statement

Show without evaluating the determinant the equality.

Homework Equations

<br /> <br /> \left(<br /> \begin{array}{ccc}<br /> 1 &amp; a &amp; bc \\<br /> 1 &amp; b &amp; ac \\<br /> 1 &amp; c &amp; ab<br /> \end{array}<br /> \right)<br /> <br />

=

<br /> <br /> \left(<br /> \begin{array}{ccc}<br /> 1 &amp; a &amp; a^2 \\<br /> 1 &amp; b &amp; b^2 \\<br /> 1 &amp; c &amp; c^2<br /> \end{array}<br /> \right)<br /> <br />

The Attempt at a Solution

I tried the facts of determinants - I did column 3 plus column 2. I can't see it. This is actually a Question on Boas 3rd Ed. Chapter 3 Section 3 No. 7.

 

[Shooting Star]

(The brackets that you are using is generally used for matrices; for a determinant, vertical lines are used to enclose it.)

Multiply the first row by 'a' and get 1/a outside. The 1st row becomes |a a^2 abc|. Now think what you can multiply the 2nd and 3rd rows with.

After that, take out something common from a column.

Then interchange columns or do a cyclic permutation on the columns, keeping track of the sign of the determinant. You will get the answer.

For any help, don't hesitate to ask.

 

[iamthegelo]

(The brackets that you are using is generally used for matrices; for a determinant, vertical lines are used to enclose it.)

Multiply the first row by 'a' and get 1/a outside. The 1st row becomes |a a^2 abc|. Now think what you can multiply the 2nd and 3rd rows with.

After that, take out something common from a column.

Then interchange columns or do a cyclic permutation on the columns, keeping track of the sign of the determinant. You will get the answer.

For any help, don't hesitate to ask.

Yeah, I actually just copied and pasted that from somewhere here that I searched for, sorry, I don't know LATEX. Thanks, I will try your suggestion.