- WELCOME TO AN EXAMPLE OF FINDING A DETERMINANT
OF A 3 X 3 MATRIX USING THE COFACTOR METHOD.
BEFORE WE START, LET'S GO OVER SOME VOCABULARY.
FIRST, THE MINOR OF ANY ELEMENT IS A DETERMINANT FORMED
BY DELETING THE ROW AND COLUMN OF THAT SPECIFIC ELEMENT.
SO FOR EXAMPLE,
IF WE CONSIDERED THE ELEMENT IN ROW 1, COLUMN 2,
OR THIS ELEMENT HERE, IT IS MINOR OR M SUB 1, 2
FOR ROW 1 COLUMN 2 IS EQUAL TO THE DETERMINANT FORMED
BY DELETING THE ROW AND COLUMN OF THIS ELEMENT.
SO IF WE ELIMINATE ROW 1 AND COLUMN 2,
NOTICE HOW IT LEAVES 4 ELEMENTS.
SO THE 2 X 2 DETERMINANT WOULD HAVE ELEMENTS DF GI.
NEXT, THE COFACTOR FOR ANY ELEMENT USING CAPITAL C
IS EQUAL TO -1 RAISED TO THE POWER OF I + J
WHERE I IS THE ROW OF THE ELEMENT
AND J IS THE COLUMN OF THE ELEMENT x ITS MINOR.
SO IF WE WANTED THE COFACTOR OF ELEMENT B
WHICH IS IN ROW 1, COLUMN 2,
WE WOULD HAVE C SUB 1, 2 IS EQUAL TO -1
RAISED TO THE POWER OF 1 + 2
WHICH WOULD BE TO THE THIRD POWER TIMES ITS MINOR
WHICH IS M SUB 1, 2.
NOW THAT WE KNOW WHAT THE MINOR AND COFACTOR ARE,
WE CAN USE THE COFACTOR METHOD
TO FIND THE DETERMINANT OF A 3 X 3 MATRIX.
TO CALCULATE THE DETERMINANT OF A 3 X 3 MATRIX,
WE MULTIPLY EACH ELEMENT IN ANY ROW OR COLUMN BY ITS COFACTOR.
THE SUM OF THESE PRODUCTS IS EQUAL TO THE DETERMINANT.
SO GOING BACK TO OUR EXAMPLE,
THE FIRST STEP IS TO DETERMINE WHICH ROW OR COLUMN
WE WANT TO USE TO FIND OUR DETERMINANT.
AND BECAUSE NONE OF THE ELEMENTS ARE 0,
IT REALLY DOES NOT MATTER,
SO LET'S GO AHEAD AND WORK OUR WAY ACROSS THIS FIRST ROW
USING THE COFACTOR METHOD.
SO THE DETERMINANT OF MATRIX A
IS GOING TO BE EQUAL TO THE FIRST ELEMENT IN ROW 1
WHICH IS 3 X -1 RAISED TO THE POWER OF THE SUM
OF THE ROW AND COLUMN OF THIS ELEMENT.
WELL, THIS ELEMENT IS IN ROW 1, COLUMN 1, 1 + 1 IS EQUAL TO 2.
WE'RE GOING TO MULTIPLY THIS
BY THE 2 X 2 DETERMINANT FORMED BY ELIMINATING
THE ROW AND COLUMN OF THE ELEMENT THREE.
SO IF WE ELIMINATE ROW 1 AND COLUMN 1,
NOTICE HOW WE ARE LEFT WITH THESE FOUR ELEMENTS HERE
FOR OUR 2 X 2 DETERMINANT.
SO WE HAVE -2, 5, 4, 1.
AND NOW WE'RE GOING TO MOVE TO THE NEXT ELEMENT IN ROW 1,
SO WE'LL HAVE PLUS THE NEXT ELEMENT IS +1 x -1
RAISED TO THE POWER OF THE ROW PLUS THE COLUMN OF THIS ELEMENT.
WELL, THIS ELEMENT IS IN ROW 1, COLUMN 2.
1 + 2 = 3.
WE'RE GOING TO MULTIPLY THIS BY THE 2 X 2 DETERMINANT
FORMED BY ELIMINATING THE ROW AND COLUMN OF THIS ELEMENT 1.
SO WE'RE GOING TO ELIMINATE ROW 1 AGAIN,
BUT NOW ELIMINATE COLUMN 2,
LEAVING US WITH THE ELEMENTS -1, 5, 2, 1
+ THE LAST ELEMENT IN THIS ROW IS +2.
WE'RE GOING TO MULTIPLY THIS BY -1
RAISED TO THE POWER OF THE ROW PLUS THE COLUMN OF THIS ELEMENT.
WELL, THE ELEMENT 2 IS IN ROW 1, COLUMN 3.
1 + 3 = 4.
AND WE'RE GOING TO MULTIPLY THIS BY THE 2 X 2 DETERMINANT
FORMED BY ELIMINATING THE ROW AND COLUMN OF THIS ELEMENT.
THIS ELEMENT 2 IS IN ROW 1, COLUMN 3,
LEAVING US WITH THE 2 X 2 DETERMINANT
WITH ELEMENTS -1, -2, 2, 4.
NOW WE'LL SIMPLIFY.
WELL, -1 SQUARED IS GOING TO BE +1.
+1 x 3 IS 3.
NOW, TO EVALUATE THIS 2 x 2 DETERMINANT
WE WOULD HAVE -2 x 1 - 5 x 4.
WELL, -2 x 1 IS -2 - 20 WOULD BE -22.
LOOKING AT THE NEXT PRODUCT WE HAVE -1 TO THE THIRD,
THAT'S -1 x 1 SO THAT'S + -1
OR JUST -1 x OUR 2 X 2 DETERMINANT
WHICH, AGAIN, WOULD BE -1 x 1 - 5 x 2.
WELL, -1 x 1 IS -1 - 10 WOULD BE -11.
NEXT WE HAVE -1 TO THE FOURTH, THAT'S +1 x 2,
SO WE'LL HAVE +2 x THE VALUE OF OUR DETERMINANT
WHICH WOULD BE -1 x 4 - -2 x 2.
WELL, -1 x 4 IS -4, AND THEN - - 2 x +2.
THAT'S - -4 OR +4.
-4 + 4 WOULD BE 0, SO WE HAVE - 66 + 11 + 0.
THIS IS EQUAL TO -55.
SO IF THE DETERMINANT OF MATRIX A IS EQUAL TO -55.
REMEMBER, WE CAN ALSO REPRESENT A DETERMINANT
USING VERTICAL BARS,
SO USING THE 9 ELEMENTS OF THE ORIGINAL MATRIX
WE CAN ALSO SAY THE DETERMINANT IN THIS FORM EQUALS -55.
OKAY, WE'LL TAKE A LOOK AT A SECOND EXAMPLE
IN THE NEXT VIDEO.
I HOPE YOU FOUND THIS HELPFUL.