- WELCOME TO A VIDEO ON DETERMINANTS.
THIS VIDEO WILL SHOW HOW TO CALCULATE DETERMINANTS
FOR 2X2 AND 3X3 MATRICES.
A DETERMINANT IS A REAL NUMBER ASSOCIATED WITH A SQUARE MATRIX.
SO IF "A" IS A SQUARE MATRIX AS WE SEE HERE
THEN A DETERMINANT OF "A" CAN BE WRITTEN LIKE THIS
OR WITH VERTICAL BARS LIKE THIS AND IT'S EQUAL TO "AD" - "BC"
AND WE ACTUALLY USED THIS IN AN EARLIER VIDEO
TO DETERMINE THE INVERSE OF A 2x2 MATRIX.
SO IF THIS IS OUR MATRIX "A"
THEN WE CAN SAY THE DETERMINANT OF MATRIX "A"
WHICH CAN BE WRITTEN LIKE THIS
IS EQUAL TO -8 x 1 THAT'S -8 - -3 x 9 WHICH IS -27.
SO THIS IS -8 + 27 WHICH EQUALS 19.
SO FOR MATRIX B
THE DETERMINANT OF B CAN BE WRITTEN LIKE THIS OR LIKE THIS
AND THIS IS EQUAL TO 7 x 4 WHICH IS 28 - 9 x 2 WHICH IS 18.
SO THE DETERMINANT OF MATRIX B IS EQUAL TO 10.
SO IT'S PRETTY STRAIGHT FORWARD FOR A 2x2.
NOW THERE'S TWO METHODS TO CALCULATE THE DETERMINANT
OF A 3x3.
METHOD 1 IS CALLED THE DIAGONAL METHOD AND THIS IS HOW IT WORKS.
WE FIRST START BY COPYING THE THREE COLUMNS OF THE MATRIX
AND THEN WE ADD COLUMN 1 AND COLUMN 2 TO THE END
TO MAKE A TOTAL OF 5 COLUMNS.
SO AGAIN, THE FIRST THREE COLUMNS
ARE FROM THE ORIGINAL MATRIX
AND THEN WE RECOPY COLUMN ONE AND COLUMN TWO.
ONCE WE HAVE IT SET UP IN THIS FASHION,
WE CAN CALCULATE THE DETERMINANT OF MATRIX "A" AS FOLLOWS.
DETERMINANT "A" IS EQUAL TO THE SUM OF THE PRODUCTS
OF THE RED DIAGONALS
THAT START AT THE TOP AND MOVE TO THE RIGHT
MINUS SUM OF THE PRODUCT OF THE GREEN DIAGONALS,
WHICH START AT THE TOP RIGHT AND MOVE TO THE LEFT.
SO WE HAVE THE QUANTITY OF AEI + BFG + CDH
MINUS THE QUANTITY OF CEG, AFH, AND BDI.
LET'S GO AHEAD AND GIVE THIS A TRY.
SO IF WE WANT TO FIND THE DETERMINANT OF MATRIX "A"
WE'LL FIRST COPY THESE FIRST THREE COLUMNS
AND THEN WE'LL COPY THE FIRST TWO COLUMNS AGAIN.
SO -2, -1, 2, AND THEN 2, 1, 0.
AND WE'RE GOING TO HAVE THE SUM OF THREE PRODUCTS
MINUS THE SUM OF THREE PRODUCTS.
THE FIRST THREE PRODUCTS WILL START AT THE TOP
AND MOVE TO THE RIGHT.
SO WE HAVE 1, 2, 3.
LET'S GO AHEAD AND FIND THOSE PRODUCTS.
SO WE HAVE -1 x 1 x -1 THAT'S +2
+ 2 x 3 x 2 THAT'S 12 + -3 x -1 x 0 IS 0.
AND NOW WE'LL FORM THE OTHER DIAGONALS
THAT START AT THE TOP AND MOVE TO THE LEFT.
SO WE HAVE 1, 2, 3.
SO LET'S GO AHEAD AND FIND THOSE PRODUCTS.
WE HAVE -3 x 1 x 2 THAT'S -6 + -2 x 3 x 0 THAT'S 0
+ 2 x -1 x -1 THAT'S +2.
IT LOOKS LIKE WE HAVE 14 - -4, WHICH IS EQUAL TO 18.
THE DETERMINANT OF MATRIX "A" IS EQUAL TO 18.
AND NOW LET'S GO AHEAD AND TAKE A LOOK AT A SECOND METHOD
FOR DETERMINING A 3x3 DETERMINANT.
IT'S CALLED THE METHOD OF COFACTORS
AND THIS IS HOW IT WORKS.
TO CALCULATE THE DETERMINANT OF A 3x3 MATRIX
WE MULTIPLY EACH ELEMENT IN ANY ROW OR COLUMN BY THE DETERMINANT
FORMED BY DELETING THE ROW AND COLUMN OF THAT ELEMENT.
IF THE SUM OF THE ROW AND COLUMN OF THE ELEMENT IS EVEN
WE'RE GOING TO ADD THE PRODUCT
AND IF THE SUM OF THE ROW AND COLUMN OF THE ELEMENT IS ODD
WE WILL SUBTRACT THE PRODUCT.
SO LETS SEE HOW THIS IS GOING TO WORK.
WE COULD SELECT ANY ROW OR COLUMN TO WORK WITH
BUT LET'S JUST GO AHEAD AND STICK TO THE FIRST ROW.
SO WHAT THAT RULE IS TELLING US IS WE TAKE THIS FIRST ELEMENT -2
AND WE'RE GOING TO MULTIPLY IT BY THE DETERMINANT
FORMED BY DELETING THE ROW AND COLUMN OF THIS ELEMENT.
SO IF WE ELIMINATE ROW 1 AND COLUMN 1
WE'RE LEFT WITH THESE FOUR ELEMENTS
WHICH FORM THE 2x2 DETERMINANTS.
SO WE HAVE 1, 3, 0, -1.
AND SINCE THIS ELEMENT IS IN ROW 1 COLUMN 1
AND 1 + 1 = 2 WHICH IS EVEN WE'RE GOING TO ADD THIS PRODUCT.
NOW WE'RE GOING TO WORK OUR WAY ACROSS ROW 1.
SO WE JUST USED THIS ELEMENT NOW WE'LL USE THIS ELEMENT
WHICH IS IN ROW 1 COLUMN 2.
SO WE'LL ELIMINATE ROW 1 COLUMN 2
AND WE'RE GOING TO HAVE THE ELEMENT 2
x THE DETERMINANT THAT WE JUST FORMED SO 1, 3, AND 2, -1.
NOW THIS ELEMENT HERE IS IN ROW 1 COLUMN 2, 1 + 2 IS 3
WHICH IS ODD SO WE HAVE TO SUBTRACT THIS PRODUCT.
AND NOW WE HAVE ONE MORE ELEMENT IN ROW 1 WE HAVE TO WORK WITH
AND NOW WE'RE GOING TO HAVE -3,
THIS ELEMENT HERE,
TIMES THE DETERMINANT FORMED BY DELETING THE ROW AND COLUMN
OF THAT ELEMENT.
SO THE DETERMINATE WILL BE -1, 1, 2, 0.
AND THEN LASTLY, THIS IS IN ROW 1 COLUMN 3, 1 + 3 = 4,
OBVIOUSLY 4 IS EVEN SO WE'LL ADD THIS PRODUCT.
SO IF WE ALWAYS USE THE FIRST ROW IT'LL BE PLUS, MINUS, PLUS.
LET'S GO AHEAD AND EVALUATE THESE 2x2 DETERMINANTS
AND THEN FIND THESE PRODUCTS.
SO HERE WE'LL HAVE -2 x THIS IS GOING TO BE -1 - 0 THAT'S -1
MINUS 2 x THIS DETERMINANT WHICH IS -1 x -1 THAT'S 1
MINUS 6 THAT'S -5.
HERE WE HAVE + -3 THE SAME AS - +3 x
HERE WE HAVE -1 x 0 THAT'S 0 - 1 x 2 THAT'S -2.
LET'S GO AHEAD AND FINISH THIS OFF.
WE'RE GOING TO HAVE +2 + 10 + 6 WHICH IS EQUAL TO 18.
AND I DON'T KNOW IF YOU NOTICED
BUT THIS IS THE SAME MATRIX THAT WE HAD BEFORE
SO OF COURSE THE DETERMINANT SHOULD BE THE SAME
AND IN FACT IT IS.
SO WE HAVE TWO WAYS OF FINDING A 3x3 DETERMINANT.
DOESN'T MATTER WHICH YOU WAY USE.
I THINK YOU HAVE TO BE A LITTLE MORE CAREFUL WITH THIS METHOD
BECAUSE BASED UPON WHAT ROW YOU CHOOSE
IT'S GOING TO EFFECT WHETHER YOU ADD OR SUBTRACT
THESE DIFFERENT PRODUCTS.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.
THANK YOU FOR WATCHING.