- WELCOME TO AN EXAMPLE
ON HOW TO FIND THE INVERSE OF A 2 BY 2 MATRIX
USING AN AUGMENTED MATRIX.
WE FIRST START BY PUTTING AN--MATRIX IN THE FORM
WHERE THE LEFT SIDE IS THE GIVEN SQUARE MATRIX
AND THE RIGHT SIDE IS THE IDENTITY MATRIX.
THEN WE PERFORM MORE OPERATIONS RESULTING IN AN AUGMENTED MATRIX
WHERE THE LEFT SIDE IS IDENTITY MATRIX
AND THE RIGHT SIDE BECOMES THE INVERSE MATRIX.
SO THE MAIN IDEA IS TO CONVERT THE GIVEN MATRIX, MATRIX "A"
INTO AN IDENTITY MATRIX USING--OPERATIONS
AND ON THE RIGHT SIDE WE END UP WITH THE INVERSE
OF DIVISIONAL MATRIX.
SO BECAUSE WE'RE GIVEN A 2 BY 2 MATRIX,
THE AUGMENTED MATRIX WILL BE A 2 BY 4 MATRIX
WHERE THE FIRST TWO COLUMNS WILL COME FROM MATRIX "A."
THE THIRD AND FOURTH COLUMNS WILL COME
FROM A 2 BY 2 IDENTITY MATRIX
SO WE'D HAVE 1, 0, 0, 1.
IF WE WANT TO WE CAN DIVIDE THIS IN HALF.
NOW WE'RE GOING TO PERFORM MORE OPERATIONS
TO OBTAIN THE IDENTITY MATRIX ON THE LEFT.
SO WHEN WE GET DONE, OUR MATRIX SHOULD LOOK JUST LIKE THIS.
SO LET'S START BY OBTAINING A 0 IN THIS POSITION
AND THIS POSITION.
SO WE'RE GOING TO HAVE A 0 IN THIS POSITION.
NOTICE THE LEAST COMMON MULTIPLE OF 2 AND 4 WOULD BE 4.
AND THESE ARE ALREADY OPPOSITE SIGNS
SO WE CAN REPLACE ROW 2 WITH 2 x ROW 2 + ROW 1
TO OBTAIN A 0 IN THIS POSITION.
SO THE FIRST ROW WILL STAY THE SAME.
NOW FOR ROW 2, WE'LL HAVE 2 x -2 + 4, THAT'S 0.
2 x -3 IS -6 + 5.
THAT'S -1 2 x 0 IS 0, + 1 IS 1
AND 2 x 1 IS 2 + 0 THAT WOULD BE 2.
NOW TO OBTAIN A 0 IN THIS POSITION HERE,
NOTICE THE ELEMENT BELOW THE 5 IS -1
WHICH MEANS WE COULD OBTAIN A 0 IN THIS POSITION
IF WE REPLACE ROW 1 WITH 5 x ROW 2.
+ ROW 1.
WE ALSO WANT THIS ELEMENT HERE TO BE +1
SO LET'S GO AHEAD AND REPLACE ROW 2 WITH -1 x ROW 2.
SO WE'LL HAVE 5 x 0 + 4 THAT'S 4, 5 x -1 IS -5 + 5, THAT'S 0.
5 x 1 IS 5 + 1 IS 6 AND 5 x 2 IS 10 + 0 IS STILL 10.
AND FOR THE SECOND ROW WE'LL HAVE 0 +1, -1, -2.
AND NOW FOR THE LAST STEP,
WE WANT THIS FIRST ELEMENT HERE TO BE +1
SO WE'LL REPLACE ROW 1 WITH 1/4 x ROW 1.
SO THE SECOND ROW STAYS THE SAME.
NOW FOR ROW 1, WE'LL HAVE 1/4 x 4, THAT'S A 1,
1/4 x 0 IS 0, 1/4 x 6 WOULD BE 6/4 OR 3 HALVES.
AND 1/4 x 10 IS 10/4 OR 5 HALVES.
NOW NOTICE ON THE LEFT SIDE OF THE AUGMENTED MATRIX
FOR THIS SIDE HERE WE HAVE THE IDENTITY MATRIX
WHICH MEANS THE RIGHT SIDE IS A INVERSE,
OR THE INVERSE OF THE ORIGINAL MATRIX.
SO THE FIRST ROW WOULD BE THREE HALVES, FIVE HALVES,
AND THE SECOND ROW WOULD BE -1, -2.
AND SINCE THIS IS THE INVERSE OF MATRIX "A,"
THIS MEANS THAT "A" x "A" INVERSE
WOULD BE EQUAL TO "A" INVERSE x A
WHICH WOULD BE EQUAL TO A 2 BY 2 IDENTITY MATRIX.
OKAY, I HOPE YOU FOUND THIS EXPLANATION HELPFUL.