Determinants – The 3×3 Matrix.

Once again, we’ll take a problem from Kelley’s Algebra pg. 122-123, Problem 6.  This is a 3 x 3 Matrix, still a square matrix but the calculation of the determinants is not as straight forward as the 2 x 2 Matrix.  Both Kelley, in his other book The Humongous Book of Algebra Problems Chapter 10, and M. Bourne  on the intmath.com site discuss more commonly used pencil and paper methods using “chords” and “minors.”  This is  transplanted jargon from music to mathematics.  I had music lessons  and I can read music;  I still don’t get it.  Jargon usually makes me suspicious.  Much to Kelley’s credit, in Chapter 9 of his Algebra book, he uses a clever method of calculating the determinants that is also easy to understand.

P6 3x3 BeforeHere is the original setup for Problem 6 with 3 rows x 3 columns.  This is the setup we will also use to calculate the determinant DET in J.

Below is  how Kelley sets up the problem to calculate the 3 x 3 determinants in J. He takes the first two columns and repeats the same numbers for the fourth and fifth columns.  He sets it up in this way so that it’s easier to calculate all of the numbers with pencil and paper  methods.  (Don’t use this matrix in J for the DET, you’ll get the wrong answer.)

P6A-KelleyCross

 

 Beginning with the number in the top left hand corner (row 1 column 1),  take  each successive number on a 45 degree angle down (_2 _5 _3) and multiply them together.  I did the math in J and assigned this as variable down1.  Then, take the next  diagonal  (_4, 1, 7) and multiply those numbers together.  After that, take the third diagonal ( 9,3,2) and multiply  them together.  I did those  in J as down2 and down3.  You might notice that he added the extra columns because otherwise the 3, 7, and 2 would be left out of  the down  calculation.

P6 Down

 

 

 

 

Then,   as with the 2×2, the up calculations are subtracted from the “downs”  So, beginning with the number in the bottom left hand corner (row 3, column 1), you multiply it together with  each  successive number at a 45 degree angle up (7, _5, 9) That, I assigned as up1.  Then take the next two diagonals and multiply them the same way: 2, 1, _2  as up2 and _3, 3,_ 4  as up3.  Notice once again that with the two columns repeated, the  _2, 3, _4  are included in the up  calculations.

P6 UP

 

 

(Opps!)

 

 

Next you add  the ups together  and then add the downs together.  Then you subtract the sum of the ups from the sum of the downs.  The answer is the Determinant.

P6-UpDownAns

I recommend that you try calculating the determinant this way once.  Then, use the DET to check your work.  But, once you understand how the determinants are calculated, I don’t see any reason not to use the J Matrix Determinant DET for these calculations.  Remember to first reshape the matrix  back to the original 3 x 3 form before attempting to use the  DET.

P6-DET

What about matrices beyond 3 x 3?  Hmmm. In Algebra, x is the horizontal dimension, y is the vertical, and z is the depth.  Other dimensions?  Time?  Or maybe it’s a mystery  from out of the Twilight Zone:

(The Twilight Zone was a popular television show in the late 1950’s and early 1960’s)

 

About Richard Rollo

I am a retired Community College Instructor. I taught Political Science 1 American Government for 22 years in Southern California. I am originally from Northern Minnesota. My earliest years were spent in the living quarters of a rural Duluth Winnipeg & Pacific Railway Depot. Then my family joined the great 1950's migration to Southern California where I joined up with fellow baby boomers in overcrowded schools.
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