Find Determinant 3

[harpazo]

Let D = determinant

1. Can D be a fraction?

2. Can D be a function?

I made up the following two problems.

Problem 1:

\begin{bmatrix}1/2&(8/9)\\

(4/5)&(6/7)\end{bmatrix}

Problem 2:

\begin{bmatrix}(x + y)&50\\

1&(x^2 + y^2)\end{bmatrix}

Can we say Problems 1 and 2 are legitimate determinant questions?

 

[Dr.Peterson]

Sure.

The determinant is just a number calculated from a matrix. It can have any value, including a fraction; the matrix can contain any numbers (even complex numbers), and if it contains variables, the determinant will be a function of those variables.

 

[harpazo]

Sure.

The determinant is just a number calculated from a matrix. It can have any value, including a fraction; the matrix can contain any numbers (even complex numbers), and if it contains variables, the determinant will be a function of those variables.

I did not know that determinants can be so much fun. Dr. Peterson, is there a general formula that I can use to find the determinant of a 3 by 3 matrix?

 

[Dr.Peterson]

Check your book! Or look lower in the Wikipedia page.

There are several ways to do it, one of which is just a formula with six terms, others being procedures that are easier to remember or to carry out than the formula (and generalizable to any size matrix).

 

[harpazo]

Check your book! Or look lower in the Wikipedia page.

There are several ways to do it, one of which is just a formula with six terms, others being procedures that are easier to remember or to carry out than the formula (and generalizable to any size matrix).

Thank you. I will check the book when I get home later tonight.

 

[Otis]

… I will check the book …

Pro Tip: You can always try that, before posting.

?

 

[harpazo]

Pro Tip: You can always try that, before posting.

?

What makes you think I do not check the book? I do not understand the math jargon used in most textbooks. This is why I plan to purchase several Michael Kelly books. He writes math books for people who do not speak mathematically.

 

[Otis]

What makes you think I do not check the book? …

One reason is because you regularly start threads to ask for definitions already listed in your book or very easily looked up online.

Dr. Peterson suggested that you post words or descriptions from your materials that confuse you. I would add to that: check other sources, first. If you can't understand a word in your materials, use a search engine to see how other authors talk about it; ask here, if you can't find any understandable description.

?

 

[harpazo]

One reason is because you regularly start threads to ask for definitions already listed in your book or very easily looked up online.

Dr. Peterson suggested that you post words or descriptions from your materials that confuse you. I would add to that: check other sources, first. If you can't understand a word in your materials, use a search engine to see how other authors talk about it; ask here, if you can't find any understandable description.

?

Good idea. From now on, I will post descriptions by Sullivan and others before the actual question. I will post the exact words that led to my confusion in the lesson section of the chapter.

 

[HallsofIvy]

I did not know that determinants can be so much fun. Dr. Peterson, is there a general formula that I can use to find the determinant of a 3 by 3 matrix?

There is but it is not easy. Given an n by n array of numbers, \(\displaystyle \{a_{ij}\}\), form all products of one number from each row and column, \(\displaystyle a_{1i_1}a_{2i}a_{3a_3}\cdot\cdot\cdot a_{ni_n}\), where \(\displaystyle i_1i_2\cdot\cdot\cdot i_n\) is permutation of \(\displaystyle 123\cdot\cdot\cdot n\), multiply by 1 if it is an even permutation or -1 if it is an odd permutation, and add them all together. For example, in the 2 by 2 determinant, \(\displaystyle \left|\begin{array}{cc}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right|\) the permutations of "12" are "12" and "21" so we form the two products \(\displaystyle a_{11}a_{22}\)and \(\displaystyle a_{12}a_{21}\). "12" is an even permutation and "21" is an odd permutation so the sum is \(\displaystyle a_{11}a_{22}- a_{12}a_{21}\).

(There are n! permutations of \(\displaystyle 123\cdot\cdot\cdot n\) so determinants get awfully complicated awfully fast! A 2 by 2 determinant involves 2 products of 2 numbers each, a 3 by 3 determinant involves 3!= 6 products of 3 numbers each, a 4 by 4 determinant involves 4!= 24 products of 4 numbers each, etc.)

 

[harpazo]

There is but it is not easy. Given an n by n array of numbers, \(\displaystyle \{a_{ij}\}\), form all products of one number from each row and column, \(\displaystyle a_{1i_1}a_{2i}a_{3a_3}\cdot\cdot\cdot a_{ni_n}\), where \(\displaystyle i_1i_2\cdot\cdot\cdot i_n\) is permutation of \(\displaystyle 123\cdot\cdot\cdot n\), multiply by 1 if it is an even permutation or -1 if it is an odd permutation, and add them all together. For example, in the 2 by 2 determinant, \(\displaystyle \left|\begin{array}{cc}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right|\) the permutations of "12" are "12" and "21" so we form the two products \(\displaystyle a_{11}a_{22}\)and \(\displaystyle a_{12}a_{21}\). "12" is an even permutation and "21" is an odd permutation so the sum is \(\displaystyle a_{11}a_{22}- a_{12}a_{21}\).

(There are n! permutations of \(\displaystyle 123\cdot\cdot\cdot n\) so determinants get awfully complicated awfully fast! A 2 by 2 determinant involves 2 products of 2 numbers each, a 3 by 3 determinant involves 3!= 6 products of 3 numbers each, a 4 by 4 determinant involves 4!= 24 products of 4 numbers each, etc.)

It does get complicated.