- WELCOME TO AN EXAMPLE ON FINDING INVERSE
OF A 2 X 2 MATRIX USING AN INVERSE FORMULA.
IF MATRIX A IS DEFINED USING ELEMENTS A, B, C, AND D,
THEN THE INVERSE OF MATRIX A IS = TO 1
DIVIDED BY THE QUANTITY A x D - B x C x THE MATRIX
WHERE WE INTERCHANGE THE POSITION OF ELEMENTS A AND D
OR THESE TWO ELEMENTS HERE.
AND THEN FOR THE ELEMENTS B AND C,
THEY STAY IN THE SAME POSITION BUT WE CHANGE THE SIGN.
SO ONCE WE FIND THE INVERSE MATRIX THE ORIGINAL MATRIX A
x A INVERSE IS = TO A INVERSE x A
WHICH WOULD BE = TO THE IDENTITY MATRIX.
AND SINCE WE'RE TALKING ABOUT A 2 X 2 MATRIX
THIS WOULD BE THE 2 X 2 IDENTITY.
SO LETS LOOK AT OUR EXAMPLE.
WE WANT TO FIND THE INVERSE OF MATRIX A,
SO A INVERSE IS GOING TO BE = TO 1
DIVIDED BY THE QUANTITY A x D
OR -3 x 6 - B x C OR -2 x 4.
SO WE WOULD HAVE -3 x 6 IS -18 - -2 x 4
THAT'S -8 x THE MATRIX
WHERE WE'LL SWITCH THE POSITIONS
OF THE ELEMENTS -3 AND 6.
WE'LL PUT THE 6 HERE AND THE -3 HERE,
AND FOR THE OTHER TWO ELEMENTS WE'LL CHANGE THE SIGN.
SO THIS -2 BECOMES 2, THIS 4 BECOMES -4.
SO LET'S GO AHEAD AND SIMPLIFY HERE.
WE HAVE 1 DIVIDED BY THIS WOULD BE -18 + 8 OR -10.
SO WE HAVE 1 DIVIDED BY -10
OR -1/10 x THE MATRIX WITH ELEMENTS 6, 2, -4, -3.
LET'S GO AHEAD AND PERFORM THIS SKIT IN MULTIPLICATION.
SO WE'LL HAVE -1/10 x 6 THAT'S -6/10 OR -3/5.
AND WE'LL HAVE -1/10 x 2 THAT'S -2/10 OR -1/5.
-1/10 x -4 IS + 4/10 OR 2/5.
AND THEN FINALLY -1/10 x -3 IS +3/10.
SO NOW THAT WE FOUND THE INVERSE OF MATRIX A,
THIS MEANS THAT MATRIX A x A INVERSE
WOULD BE = TO A INVERSE x A,
WHICH IS = TO THE IDENTITY MATRIX.
IN THIS CASE WOULD BE A 2 X 2 IDENTITY MATRIX.
LET'S GO AHEAD AND CHECK THIS PRODUCT ON THE NEXT SLIDE.
AGAIN, NOW WE'LL CHECK THAT A x THE INVERSE
IS = TO A 2 X 2 IDENTITY MATRIX.
SO JUST TO VERIFY THAT WE CAN PERFORM THIS MULTIPLICATION
WE HAVE A 2 X 2 MATRIX HERE, A 2 X 2 MATRIX HERE.
SO BECAUSE THE NUMBER OF COLUMNS IN THE FIRST MATRIX
EQUALS THE NUMBER OF ROWS IN THE SECOND MATRIX,
THE MULTIPLICATION IS DEFINED,
AND THEN NUMBER OF ROWS IN THE FIRST MATRIX,
NUMBER OF COLUMNS IN THE SECOND MATRIX
GIVE US THE DIMENSIONS OF OUR PRODUCT.
SO OUR PRODUCT IS A 2 X 2 MATRIX.
AND NOW TO FIND THIS PRODUCT WE'LL BE USING ROWS
FROM THE FIRST MATRIX
AND COLUMNS FROM THE SECOND MATRIX.
SO TO FIND THE ELEMENT IN ROW ONE, COLUMN ONE,
OR THIS ELEMENT HERE, IT SHOULD BE A SUB 1, 1,
WE'RE GOING TO MULTIPLY ROW ONE BY COLUMN ONE.
WHICH MEANS WE'LL HAVE -3 x -3/5 OR -3/1 x -3/5 + -2
OR -2/1 x 2/5.
SO HERE WE'LL HAVE, WHAT, 9/5 + -4/5,
WHICH IS = TO 5/5 OR 1.
NEXT WE'LL FIND THE ELEMENT IN ROW ONE, COLUMN TWO.
SO WE USE ROW ONE FROM THE FIRST MATRIX
BUT NOW COLUMN TWO FROM THE SECOND MATRIX.
SO WE'LL HAVE -3 OR -3/1 x -1/5 + -2/1 x 3/10.
SO HERE WE'LL HAVE +3/5
+ NOTICE WE HAVE A COMMON FACTOR OF 2 HERE,
THIS SIMPLIFIES TO 1, THIS SIMPLIFIES TO 5.
SO WE HAVE + THIS WILL BE -3/5, WHICH IS = TO 0.
NOW WE'LL FIND THE ELEMENT IN ROW TWO, COLUMN ONE.
SO NOW WE'LL USE ROW TWO FROM THE FIRST MATRIX
AND COLUMN ONE FROM THE SECOND MATRIX.
WE'LL HAVE 4 x -3/5 OR 4/1 x -3/5 + 6 OR 6/1 x 2/5.
SO WE'LL HAVE -12/5 + 12/5 WHICH IS = TO 0.
SO FAR SO GOOD.
AND NOW FOR THE LAST ELEMENT IN ROW TWO, COLUMN TWO,
WE'LL USE ROW TWO FROM THE FIRST MATRIX
AND COLUMN TWO FROM THE SECOND MATRIX.
WE HAVE 4 x -1/5 OR 4/1 x -1/5 + 6 OR 6/1 x 3/10.
SO WE HAVE -4/5 + AGAIN, THIS SIMPLIFIES,
WE HAVE A COMMON FACTOR OF 2.
THERE ARE FIVE 2's IN 10 AND THREE 2's IN 6.
THIS WILL BE 9/5, WHICH IS = TO 5/5, WHICH IS = TO 1.
SO WE JUST VERIFIED THE GIVEN MATRIX A
TIMES THE INVERSE MATRIX
THAT WE JUST FOUND IS EQUAL TO A 2 X 2 IDENTITY MATRIX.
OKAY. I HOPE YOU FOUND THIS EXPLANATION HELPFUL.