- WELCOME TO AN EXAMPLE ON HOW TO FIND THE INVERSE
OF A 3 X 3 MATRIX USING AN AUGMENTED MATRIX.
THE FIRST STEP IS TO CREATE AN AUGMENTED MATRIX IN THE FORM
WHERE THE LEFT SIDE IS MATRIX A OR THE GIVEN MATRIX,
AND THE RIGHT SIDE IS AN IDENTITY MATRIX.
AND WE PERFORM ROW OPERATIONS
RESULTING IN AN AUGMENTED MATRIX IN THE FORM
WHERE THE LEFT SIDE BECOMES THE IDENTITY MATRIX,
AND THE RIGHT SIDE WILL BE THE INVERSE OF MATRIX A.
SO THE MAIN IDEA IS TO SET UP THE AUGMENTED MATRIX
WHERE MATRIX A IS ON THE LEFT SIDE OF THE MATRIX,
PERFORM ROW OPERATIONS TO FORM THE IDENTITY MATRIX
FROM MATRIX A,
AND THE RIGHT SIDE WILL BE A INVERSE
OR THE INVERSE OF THE ORIGINAL MATRIX.
SO BECAUSE WE'RE GIVEN A 3 X 3 MATRIX,
WE'LL HAVE THE GIVEN MATRIX ON THE LEFT,
AND ON THE RIGHT SIDE WE'LL HAVE A 3 X 3 IDENTITY MATRIX.
SO WE'LL HAVE A 3 X 6 AUGMENTED MATRIX
WHERE THE FIRST THREE COLUMNS WILL COME FROM MATRIX A,
AND THE FOURTH, FIFTH, AND SIXTH COLUMNS
WILL COME FROM A 3 X 3 IDENTITY MATRIX.
SO WE WOULD HAVE 1, 0, 0, 0, 1, 0, AND 0, 0, 1.
A LOT OF TIMES YOU'LL SEE THE AUGMENTED MATRIX
DIVIDED IN HALF.
NOW WE'LL PERFORM ROW OPERATIONS
SO THE AUGMENTED MATRIX HAS A 3 X 3 IDENTITY MATRIX
ON THE LEFT
OR LOOKS JUST LIKE THIS MATRIX HERE.
IN THIS FORM THE THREE COLUMNS ON THE LEFT
WOULD BE THE INVERSE OF MATRIX A.
SO WE'LL START BY OBTAINING 0's IN THE CORRECT POSITIONS,
AND THEN MAKE SURE THIS DIAGONAL HERE
CONSIST ONLY OF 1's.
LET'S START BY OBTAINING A 0 IN THIS POSITION HERE.
NOTICE THE ELEMENT ABOVE IT IS -2.
SO IF THIS WAS 4,
WE COULD THEN ADD ROW TWO AND THREE TO OBTAIN A 0.
SO LET'S REPLACE ROW 3 WITH -2 x ROW 2 + ROW 3.
LET'S ALSO OBTAIN A 0 IN THIS POSITION HERE
USING ROW ONE AND ROW TWO.
THE LEAST COMMON MULTIPLE OF 2 AND 3 IS 6,
SO IF WE MULTIPLY ROW ONE BY TWO,
THEN ADD IT TO 3 x ROW 2 WE WOULD HAVE A 0 HERE.
SO 2 x ROW 1 + 3 X ROW 2.
AND THEN WHILE WE'RE HERE
LET'S GO AHEAD AND OBTAIN A 0 IN THIS POSITION HERE.
NOTICE THE ELEMENT BELOW IT IS 1,
SO LET'S GO AHEAD AND REPLACE ROW ONE
WITH ROW ONE + ROW TWO.
NOTICE WE'RE PERFORMING THREE DIFFERENT ROW OPERATIONS
ALL IN ONE STEP.
SO STARTING IN THE FIRST ROW WE'LL ADD ROW ONE AND ROW TWO.
SO 3 + -2 IS 1, 3 + -2 IS 1 AGAIN.
-1 + 1 IS 0, 1 + 0 IS 1, 0 + 1 IS 1, AND 0 + 0 IS 0.
WE NEED TO BE EXTRA CAREFUL ON THE SECOND OPERATION.
WE HAVE 2 x ROW 1 + 3 x ROW 2.
SO 2 x 3 IS 6 + 3 x -2 THAT'S -6, SO THAT'S 0.
2 x 3 IS 6 + 3 x -2 IS -6 THAT'S ALSO 0.
NEXT WE HAVE 2 x -1 THAT'S -2 + 3 x 1 THAT'S 3,
SO THAT'S 1.
NEXT WE HAVE 2 x 1 THAT'S 2 + 3 x 0 THAT'S STILL 2.
2 x 0 IS 0 + 3 x 1 IS 3, SO THAT'S 3.
2 x 0 IS 0 + 3 x 0 IS STILL 0.
FOR ROW THREE WE HAVE -2 x ROW 2 + ROW 3.
SO -2 x -2 IS 4 + -4 THAT'S 0.
-2 x -2 IS 4 + -5 THAT'S -1.
-2 X 1 THAT'S -2 + 2 THAT'S 0.
AND THEN WE HAVE -2 x 0 THAT'S 0 + 0, STILL 0.
-2 x 1 THAT'S -2 + 0 THAT'S -2, AND -2 x 0 + 1 IS 1.
WE MADE A LOT OF GOOD PROGRESS SO FAR.
NOTICE WE COULD OBTAIN A 0 IN THIS POSITION HERE
IF WE REPLACE ROW 1 WITH ROW 1 + ROW 3.
LET'S GO AHEAD AND DO THAT.
BUT NOTICE HOW ROW TWO HAS A IN COLUMN ONE AND COLUMN TWO,
WHICH WOULD BE BETTER IF IT WAS IN ROW THREE.
SO LET'S GO AHEAD AND INTERCHANGE ROW TWO
AND ROW THREE.
WELL, FOR ROW ONE WE HAVE 1 + 0 THAT'S 1,
1 + -1 THAT'S 0, 0 + 0 IS 0, 1 + 0 IS 1, 1 + -2 IS -1,
AND FINALLY 0 + 1 IS 1.
AND THEN WE'RE GOING TO INTERCHANGE ROW TWO
AND ROW THREE,
SO WE HAVE 0, - 1, 0, 0, -2, 1, AND 0, 0, 1, 2, 3, 0.
NOTICE HOW WE'RE ALMOST THERE NOW.
THIS FIRST ELEMENT IN ROW TWO HAS TO BE +1,
SO FOR OUR LAST STEP WE'RE GOING TO REPLACE ROW TWO
WITH -1 x ROW 2.
ROW ONE AND ROW THREE STAY THE SAME.
AND THEN MULTIPLYING BY -1 IS GOING TO CHANGE THE SIGN
OF ALL THE ELEMENTS IN ROW TWO,
SO WE'LL HAVE 0, 1, 0, 0, 2, -1.
NOW NOTICE THIS AUGMENTED MATRIX IS IN THE FORM
THAT WE NEED WHERE THE LEFT SIDE OF THE AUGMENTED MATRIX
IS A 3 X 3 IDENTITY MATRIX,
WHICH MEANS THE RIGHT SIDE IS A INVERSE
OF THE INVERSE OF THE ORIGINAL 3 X 3 MATRIX.
SO A INVERSE IS = TO A 3 X 3 MATRIX
WHERE ROW ONE IS 1, -1, 1.
ROW TWO IS 0, 2, -1, AND ROW THREE IS 2, 3, 0.
WHICH MEANS THAT A x A INVERSE IS = TO A INVERSE x A,
WHICH WOULD BE = TO A 3 X 3 IDENTITY MATRIX.
OKAY. I HOPE YOU FOUND THIS EXPLANATION HELPFUL.
WE'LL TAKE A LOOK AT ANOTHER EXAMPLE IN THE NEXT VIDEO.