- WELCOME TO A VIDEO ON DETERMINING
THE INVERSE OF A MATRIX USING AUGMENTED MATRICES.
IF "A" IS AN N BY N INVERTIBLE MATRIX
SUCH THAT "A" x B = B x "A" WHICH EQUALS THE IDENTITY MATRIX
THEN B IS THE INVERSE OF "A"
DENOTED WITH THIS INVERSE NOTATION.
SO LET'S GO AHEAD AND TAKE A LOOK
AT WHAT HAPPENS WHEN WE FIND THE PRODUCT OF THESE TWO MATRICES.
FIRST WE HAVE "A" x B AND THEN WE HAVE B x "A."
NOW REMEMBER USUALLY MATRIX MULTIPLICATION
IS NOT COMMUTATIVE BUT LET'S GO AHEAD AND SEE
WHAT HAPPENS IN THIS CASE
AND WE'RE GOING TO USE THE CALCULATOR
BECAUSE WE'VE ALREADY COVERED HOW TO MULTIPLY BY HAND.
SO WE'RE GOING TO PRESS 2nd MATRIX, GO OVER TO EDIT,
PRESS ENTER, AND GO AHEAD AND ENTER IN MATRIX "A,"
IT'S A 2 BY 2 AND THEN WE HAVE THE ELEMENTS 4, 7, 1, AND 2.
LET'S GO AHEAD AND ENTER IN MATRIX B.
2nd MATRIX, GO OVER TO EDIT AND THEN DOWN TO MATRIX B,
AGAIN IT'S A 2 BY 2
AND THE ELEMENTS ARE 2, -7, -1, AND 4.
LET'S GO AHEAD AND FIND THOSE TWO PRODUCTS NOW.
LET'S GO BACK TO THE MAIN SCREEN
AND THEN PRESS 2nd MATRIX "A" TIMES 2nd MATRIX, MATRIX B
AND THEN PRESS ENTER AND WE CAN SEE THAT "A" x B
IS A 2 BY 2 IDENTITY MATRIX AND NOW LET'S FIND B x "A."
AND YOU NOTICE THAT THE PRODUCT IS THE SAME.
SO SINCE THE PRODUCT OF THESE TWO MATRICES
IS THE IDENTITY MATRIX WE CAN SAY B IS THE INVERSE OF "A."
LET'S GO AHEAD AND WRITE THIS OUT.
SO WHAT THAT MEANS IS THAT MATRIX B
IS ACTUALLY THE INVERSE OF MATRIX "A."
SO NOW WHAT WE'LL DO IS TAKE A LOOK AT HOW
WE FIND THE INVERSE OF THE GIVEN MATRIX
AND WE'RE GOING TO USE AUGMENTED MATRICES TO DO THIS.
SO WHAT WE'RE GOING TO DO IS WE'RE GOING TO CREATE
AN AUGMENTED MATRIX IN THE FORM OF MATRIX "A" ON THE LEFT
AND THE IDENTITY MATRIX ON THE RIGHT.
SO WHAT WE'LL DO THEN IS PERFORM ROW MATRIX OPERATIONS
THAT WILL RESULT IN THE IDENTITY MATRIX BEING ON THE LEFT SIDE
AND THE RIGHT SIDE WILL BECOME THE INVERSE OF MATRIX "A."
SO THE MAIN IDEA HERE IS TO CONVERT MATRIX "A"
ON THE LEFT INTO THE IDENTITY MATRIX
AND THEN THE RIGHT SIDE WILL BE THE INVERSE OF MATRIX "A."
LET'S GO AHEAD AND GIVE IT A TRY.
SO IF WE WANT TO FIND THE INVERSE OF THIS MATRIX
WE'LL SET UP AN AUGMENTED MATRIX
WHERE THE LEFT SIDE WILL BE THE GIVEN MATRIX
AND THE RIGHT SIDE WILL BE THE IDENTITY MATRIX.
AND SINCE THIS IS A 2 BY 2 MATRIX
WE WANT TO MAKE THE LEFT SIDE LOOK LIKE THE RIGHT SIDE.
SO WE WANT A 1 HERE, A 1 HERE,
AND WE WANT THESE TWO ELEMENTS TO BE ZERO.
SO IF WE WANT TO HAVE A ZERO IN THIS LOCATION
THE CORRESPONDING ELEMENT WOULD BE A 3.
THE LEAST COMMON MULTIPLE OF 4 AND 3 WOULD BE 12.
SO WHAT WE COULD DO IS MULTIPLY THIS ROW BY 3,
MULTIPLY THIS ROW BY 4 AND THEN SUBTRACT THEM,
SO WE'RE GOING TO REPLACE ROW 2 WITH 3 x ROW 2 - 4 x ROW 1.
AND IF WE WANT A ZERO IN THIS LOCATION
THE CORRESPONDING ELEMENT IS 3,
THE LEAST COMMON MULTIPLE OF 2 AND 3 WOULD BE 6.
SO WE'RE GOING TO MULTIPLY ROW 1 BY 3,
MULTIPLY ROW 2 BY 2 AND THEN SUBTRACT
SO WE'LL REPLACE ROW 1 WITH 3 x ROW 1 - 2 x ROW 2.
LET'S GO AHEAD AND DO THAT AND SEE WHAT HAPPENS.
SO WE'RE GOING TO HAVE 3 x 3 - 2 x 4 THAT'S 9 - 8 THAT'S 1.
HERE WE'RE GOING TO HAVE 3 x 2 THAT'S 6 - 2 x 3
WHICH IS ALSO 6.
6 - 6 WOULD BE 0 AND HERE WE'D HAVE 3 x 1 - 2 x 0
THAT WOULD BE 3, AND HERE WE'D HAVE 3 x 0
WHICH IS 0 - 2 x 1 THAT'S -2.
NOW LETS TAKE A LOOK AT ROW 2.
WE'RE GOING TO HAVE 3 x 4 - 4 x 3 THAT'S 0
AND HERE WE'RE GOING TO HAVE 3 x 3 THAT'S 9 - 4 x 2 THAT'S 8.
9 - 8 WOULD BE 1 AND HERE WE HAVE 3 x 0 - 4 x 1 THAT'S -4
AND THEN LASTLY, WE HAVE 3 x 1 THAT WOULD BE 3 - 4 x 0
IT'S STILL 3.
SO NOTICE NOW THE LEFT SIDE IS THE IDENTITY MATRIX,
WHICH MEANS NOW THE RIGHT SIDE
SHOULD BE THE INVERSE OF MATRIX "A."
LET'S GO AHEAD AND WRITE THIS DOWN
AND THEN WE'LL TEST IT ON THE CALCULATOR.
SO "A" INVERSE SHOULD BE 3, -2, -4, 3.
REMEMBER FROM THE PREVIOUS SCREEN,
THIS MEANS THAT "A" x "A" INVERSE
SHOULD EQUAL THE IDENTITY MATRIX AND "A" INVERSE x "A"
SHOULD ALSO EQUAL THE IDENTITY MATRIX.
SO LET'S GO AHEAD AND CHECK THIS.
GO AHEAD AND ENTER IN THE GIVEN MATRICES.
SO 2nd MATRIX, GO OVER TO EDIT, PRESS ENTER,
AND WE'RE GOING TO ENTER IN MATRIX "A" HERE.
3, 2, 4, 3, AND THEN WE'LL PUT "A" INVERSE IN MATRIX B.
OKAY, LET'S GO AHEAD AND FIND THESE TWO PRODUCTS.
SO MATRIX "A" x MATRIX B AND THAT CHECKS OUT.
NOW LET'S GO AHEAD AND FIND MATRIX B TIMES MATRIX "A."
AND WE HAVE FOUND THE INVERSE CORRECTLY.
LET'S GO AHEAD AND SEE IF WE CAN FIND A 3 BY 3 INVERSE MATRIX.
SO NOW WE'RE GOING TO HAVE A 3 BY 6 AUGMENTED MATRIX.
SO WE HAVE MATRIX "A" ON THE LEFT
AND WE'LL HAVE THE IDENTITY MATRIX ON THE RIGHT.
SO WE WANT THE MATRIX ON THE LEFT
TO BECOME THE IDENTITY MATRIX
SO WE WANT THESE THREE ELEMENTS TO BE ZERO,
THESE THREE TO BE ZERO,
AND THEN ALL OF THESE TO BE
AND THEN THE MAIN DIAGONAL TO BE ONE'S.
SO IF WE WANT THIS ELEMENT HERE TO BE A ZERO
WE CAN ADD IT TO ROW 2.
SO LET'S GO AHEAD AND REPLACE
ROW 1 WITH ROW 1 + ROW 2
AND IF WE WANT THIS ELEMENT HERE TO BE A ZERO
WE COULD MULTIPLY IT BY 2 AND THEN ADD IT TO ROW 1.
SO LET'S REPLACE ROW 2 WITH 2 x ROW 2 + ROW 1.
SO ADDING ROW 1 AND ROW 2 WE'LL HAVE 1, 0, 1, 1, 1, AND 0.
AND NOW, MULTIPLY ROW 2 AND THEN ADD IT TO ROW 1.
2 x 0 + 1 THAT'S 1.
2 x 1 IS 2 + -1 THAT'S 1.
HERE WE HAVE 2 x -1 THAT'S -2 + 2 THERE'S 0 THERE.
2 x 0 + 1, 2 x 1 + 0 THAT'S 2,
AND 2 x 0 + 0 WOULD STILL BE 0 AND WE LEFT ROW 3 THE SAME.
NOW IF WE WANT TO GET A ZERO HERE
WE'RE GOING TO HAVE TO WORK WITH ROW 3
OTHERWISE WE WOULD LOSE THIS ZERO HERE.
SO LET'S MULTIPLY ROW 1 BY 2 AND THEN SUBTRACT ROW 3.
SO 2 x ROW 1 - ROW 3 WOULD GIVE US A ZERO HERE
AND THEN GOING DOWN TO ROW 3 IF WE TAKE ROW 3
AND SUBTRACT ROW 1 WE'LL GET A ZERO HERE
WITHOUT LOSING THIS ZERO HERE.
LET'S GO AHEAD AND DO THAT.
ROW 3 - ROW 1, SO 2 x 1 WOULD BE 2 - 1 THAT'S 1.
2 x 0 WOULD BE 0 - 0, STILL 0.
2 x 1 IS 2 - 2 THERE'S A 0.
2 x 1 THAT WOULD BE 2 - 0 THAT WOULD BE 2.
2 x 1 WOULD BE 2 - 0 AND 2 x 0 THAT'S 0 - 1 SO WE HAVE -1.
THE SECOND ROW STAYS THE SAME.
THE THIRD ROW IS ROW 3 - ROW 1.
SO 1 - 1, 0, 0 - 0, 2 - 1, 1.
0 - 1, -1, 0 - 1 THAT'S -1 AND THEN 1 - 0 IS 1.
OKAY, WE'RE CLOSE.
YOU CAN SEE WE HAVE A MAIN DIAGONAL OF ONE'S.
WE HAVE ZEROS HERE.
WE NEED THIS ELEMENT HERE TO BE ZERO.
SO WE'LL REPLACE ROW 2 WITH ROW 2 - ROW 1.
WE'RE OUT OF SPACE HERE SO WHAT WE'RE GOING TO DO
IS SEE IF WE CAN SQUEEZE IT IN OVER HERE ON THE LEFT.
ROW 1 AND ROW 3 STAY THE SAME.
SO AGAIN, WE HAVE ROW 2 - ROW 1 SO WE HAVE 1 - 1 THAT'S 0.
1 - 0, THAT'S 1.
0 - 0 IS 0.
1 - 2 THAT'S GOING TO BE -1.
2 - 2 THAT WOULD BE 0 AND THEN 0 - -1 BECOMES A 1.
SO AGAIN, THE LEFT SIDE IS NOW THE IDENTITY MATRIX.
THE RIGHT SIDE SHOULD BE "A" INVERSE.
2, 2, -1, -1, 0, 1, AND -1, -1, AND 1.
WE'RE A LITTLE BIT SHORT ON TIME
SO YOU MAY WANT TO GO AHEAD AND TEST TO MAKE SURE
THAT "A" x "A" INVERSE AND "A" INVERSE x "A"
DOES EQUAL THE IDENTITY MATRIX.
I'VE ALREADY DONE THAT AND IT DOES CHECK OUT.