- WELCOME TO A SECOND EXAMPLE ON HOW TO FIND THE INVERSE
OF A 3 X 3 MATRIX USING AN AUGMENTED MATRIX.
BECAUSE THE GIVEN MATRIX IS A 3 X 3 MATRIX,
OUR AUGMENTED MATRIX WILL BE A 3 X 6 MATRIX
WHERE ON THE LEFT WE'LL HAVE MATRIX A,
AND ON THE RIGHT WE'LL HAVE A 3 X 3 IDENTITY MATRIX.
SO LET'S GO AHEAD AND SET THIS UP.
AGAIN, WE HAVE MATRIX A ON THE LEFT
AND A 3 X 3 IDENTITY MATRIX ON THE RIGHT.
NOW WE'RE GOING TO PERFORM ROW OPERATIONS
TO TRANSFORM THE LEFT SIDE
OR MATRIX A INTO A 3 X 3 IDENTITY MATRIX
AS WE SEE HERE IN RED.
AND ONCE WE HAVE A 3 X 3 IDENTITY MATRIX
ON THE LEFT HERE,
THE RIGHT SIDE WILL BE THE INVERSE OF MATRIX A
OR A INVERSE.
SO WE NEED TO START BY OBTAINING ZEROS
IN THE SIX POSITIONS,
AND THEN MAKE SURE THE MAIN DIAGONAL HERE
CONSIST ONLY OF ONES.
SO LOOKING AT OUR MATRIX,
LET'S START BY OBTAINING A ZERO HERE AND HERE.
WELL, IF WE MULTIPLY ROW 2 BY 3
AND THEN ADDED IT TO ROW 3 WE'D HAVE A 0
IN THIS POSITION HERE.
SO LET'S GO AHEAD AND REPLACE ROW 3 WITH 3 x ROW 2 + ROW 3.
NOW, LET'S ALSO OBTAIN A ZERO IN ROW TWO, COLUMN ONE,
AND LET'S FOCUS ON USING ROW ONE AND ROW TWO.
IF WE REPLACE ROW 2 WITH -4 x ROW 2 + ROW 1
THAT WOULD GIVE US A 0 IN THIS FIRST POSITION.
NOW, IF WE LOOK AT ROW ONE, COLUMN THREE
OR THIS ELEMENT HERE,
WE COULD OBTAIN A ZERO HERE
ALSO IF YOU REPLACE ROW ONE WITH ROW ONE + ROW THREE.
NOTICE HOW WE HAVE A ONE IN THIS POSITION HERE.
SO AT THE SAME TIME LET'S GO AHEAD AND REPLACE ROW ONE
WITH ROW ONE + ROW THREE.
SO FOR ROW ONE WE'LL HAVE 4 + -3 THAT'S 1,
2 + -1 THAT'S 1, -1 + 1 IS 0.
THEN WE HAVE 1 + 0 IS 1, 0 + 0 IS 0, AND 0 + 1 IS 1.
NOW FOR ROW TWO WE'LL HAVE -4 x 1 + 4 THAT'S 0,
AND THEN -4 x 1 THAT'S -4 + 2 THAT'S -2, -4 x -1
THAT'S +4 + -1 THAT'S 3.
AND THEN WE HAVE -4 x 0 + 1 IS 1,
-4 x 1 THAT'S -4 + 0 IS -4, AND -4 x 0 + 0 IS 0.
FINALLY FOR ROW THREE WE HAVE 3 x ROW 2 + ROW 3.
SO FIRST WE HAVE 3 x 1 THAT'S 3 + -3 THAT'S 0,
3 x 1 THAT'S 3 + -1 THAT'S 2.
3 x -1 THAT'S -3 + 1 THAT'S -2, 3 x 0 + 0 IS 0,
3 x 1 IS 3 + 0 IS 3.
AND 3 x 0 IS 0 + 1 IS 1.
NOW LET'S TRY TO GET A ZERO HERE AND HERE,
AND WE'LL HAVE TO USE ROW TWO AND THREE
SO WE DON'T LOSE THE ZERO IN COLUMN ONE.
SO NOTICE THAT IN COLUMN TWO
THESE TWO ELEMENTS ARE ALREADY OPPOSITES,
SO LET'S GO AHEAD AND REPLACE ROW THREE
WITH ROW TWO + ROW THREE.
NOW, LOOKING AT ROW TWO, COLUMN THREE,
NOTICE THE ELEMENT BELOW THE THREE IS ALREADY NEGATIVE,
SO THEY'RE OPPOSITE SIGNS.
AND SINCE THE LEAST COMMON MULTIPLE OF 2 AND 3 IS 6
WE'RE GOING TO REPLACE ROW 2 WITH 2 x ROW 2 + 3 x ROW 3.
SO WE'LL KEEP THE FIRST ROW THE SAME FOR RIGHT NOW.
AND NOW FOR 2 x ROW 2 + 3 x ROW 3
WE'LL HAVE 2 x 0 + 3 x 0 THAT'S 0.
2 x -2 THAT'S -4 + 3 x 2 THAT'S +6, SO THAT'S +2.
2 x 3 + 3 x -2 THAT'S 0.
AND WE HAVE 2 X 1 + 3 x 0 THAT'S 2.
THEN 2 x -4 THAT'S -8 + 3 x 3 THAT'S +9 THAT'S +1.
3 x 0 + 3 x 1 IS 3.
FOR ROW THREE WE HAVE ROW TWO + ROW THREE.
SO WE HAVE 0 + 0 THAT'S 0, 2 + -2 THAT'S 0, -2 + 3 IS 1,
0 + 1 IS 1, 3 + -4 IS -1, AND 1 + 0 IS 1.
OKAY. SO NOW WE'RE GETTING CLOSE.
WE STILL NEED TO HAVE A ZERO IN THIS POSITION HERE,
SO WE'LL GO AHEAD AND USE ROW ONE AND ROW TWO.
WE NEED THIS ELEMENT HERE TO BE -2 SO WE CAN ADD IT TO +2.
SO WE'RE GOING TO REPLACE ROW 1 WITH -2 x ROW 1 + ROW 2.
WE ALSO WANT THIS FIRST ELEMENT IN ROW TWO TO BE +1,
SO WE'LL ALSO REPLACE ROW TWO WITH 1/2 x ROW 2.
SO -2 x 1 + 0 IS -2.
-2 x 1 + 2 IS 0, -2 x 0 + 0 IS 0,
-2 x 1 IS -2 + 2 IS 0, -2 x 0 + 1 IS 1, -2 x 1
THAT'S -2 + 3 IS +1.
AND THEN FOR ROW TWO WE'RE GOING TO MULTIPLY BY 1/2,
SO WE HAVE 0, 1, 0, 1, 1/2, AND 3/2.
ROW THREE STAYS THE SAME.
NOW FOR THE LAST STEP,
THIS FIRST ELEMENT HERE HAS TO BE +1,
SO WE'RE GOING TO REPLACE ROW ONE WITH -1/2 x ROW 1.
SO ROW TWO AND ROW THREE STAY THE SAME.
SO WE HAVE -1/2 X -2 THAT'S +1,
AND THEN -1/2 x 0 IS 0, SO WE HAVE 0, 0, 0.
AND WE HAVE -1/2, -1/2.
NOTICE NOW WE DO HAVE THE FORM THAT WE NEED,
SO THE LEFT SIDE IS A 3 X 3 IDENTITY MATRIX,
WHICH MEANS THE RIGHT SIDE IS A INVERSE.
SO A INVERSE OR THE INVERSE OF MATRIX A
IS EQUAL TO THE 3 X 3 MATRIX
WHERE THE FIRST ROW IS 0, - 1/2, -1/2.
THE SECOND ROW IS 1, 1/2, 3/2.
AND THE THIRD ROW IS 1, -1, 1.
AND NOW THAT WE HAVE THE INVERSE WE KNOW
THAT A x A INVERSE = A INVERSE x A,
WHICH IS = TO A 3 X 3 IDENTITY MATRIX.
I HOPE YOU FOUND THIS HELPFUL.