- WELCOME TO AN EXAMPLE ON HOW TO SOLVE A SYSTEM
OF THREE LINEAR EQUATIONS
WITH THREE UNKNOWNS USING A MATRIX EQUATION.
BEFORE WE SOLVE THIS SYSTEM,
LET'S REVIEW THE PROCEDURE FOR USING MATRIX EQUATION.
THE FIRST STEP IS TO MAKE SURE THE LINEAR EQUATIONS
ARE IN STANDARD FORM,
MEANING WE HAVE THE X TERMS, THE Y TERMS,
AND THE Z TERMS ON THE LEFT SIDE OF THE EQUATION,
AND THE CONSTANTS ON THE RIGHT.
ONCE THE EQUATIONS ARE IN THE CORRECT FORM,
WE'RE GOING TO FORM THREE MATRICES,
MATRIX A WILL BE THE COEFFICIENT MATRIX,
MATRIX X WILL BE THE VARIABLE MATRIX,
AND MATRIX B WILL BE THE CONSTANT MATRIX.
THE ELEMENTS IN MATRIX A
WILL BE THE COEFFICIENTS OF THE X, Y, AND Z TERMS.
THE VARIABLE MATRIX X WILL CONTAIN THE VARIABLES
USED IN THE SYSTEM,
AND MATRIX B WILL BE THE CONSTANTS,
OR THE VALUES ON THE RIGHT SIDE OF EACH EQUATION.
ONCE WE HAVE THESE THREE MATRICES,
THE MATRIX EQUATION
WILL BE MATRIX A x MATRIX X = MATRIX B.
ONCE WE HAVE IT IN THIS FORM,
OUR GOAL IS TO SOLVE THEEQUATION FOR MATRIX X.
TO DO THIS, WE'LL FIND THE INVERSE OF MATRIX A
AND THEN MULTIPLY BOTH SIDES OF THE EQUATION
BY A INVERSE HERE AND HERE.
AND SINCE MATRIX MULTIPLICATION IS NOT COMMUNICATIVE,
THIS ORDER IS IMPORTANT.
WE HAVE TO HAVE A INVERSE x A ON THE LEFT
AND A INVERSE x B ON THE RIGHT.
THEN WE SHOULD RECOGNIZE
THAT A INVERSE x A IS EQUAL TO THE IDENTITY MATRIX,
AND THE IDENTITY MATRIX x MATRIX X WHICH IS B MATRIX X.
SO LOOKING AT THE RIGHT SIDE OF THE EQUATION A INVERSE x B
WILL BE THE SOLUTION TO OUR SYSTEM.
SO GOING BACK TO OUR EXAMPLE,
NOTICE AGAIN, ALL THE EQUATIONS ARE IN STANDARD FORM.
SO THIS WILL BE MATRIX A, THIS WILL BE MATRIX X,
AND THIS WILL BE MATRIX B.
SO LOOKING AT THE COEFFICIENTS THE FIRST ROW IN MATRIX A
WILL BE 1,2,-1.
THE SECOND ROW WILL BE 2,-3,-4,
AND THE THIRD ROW WILL BE 1,1,1.
NEXT, MATRIX X
BECAUSE WE'RE USING THE VARIABLES X, Y, AND Z
WILL JUST BE X, Y, Z,
AND MATRIX B WILL CONTAIN 7, -3, 0.
AGAIN, OUR GOAL HERE IS TO SOLVE FOR MATRIX X,
SO OUR NEXT STEP IS TO FIND THE INVERSE OF MATRIX A
AND THEN MULTIPLY BOTH SIDES OF THE EQUATION BY A INVERSE.
SO WE WOULD HAVE A INVERSE HERE
x MATRIX A x MATRIX X = A INVERSE x MATRIX B.
LET'S GO AHEAD AND FILL IN MATRIX A, X, AND B,
AND THEN WE'LL FIND THE INVERSEMATRIX.
NOW, FOR THIS VIDEO
WE'RE NOT GOING TO FIND THE INVERSE OF THE 3x3
MATRIX BY HAND,
BUT I WILL PUT LINKS ON THE SCREEN TO VIDEOS THAT WILL SHOW
HOW TO FIND THE INVERSE OF A 3x3 MATRIX.
WE'RE GOING TO GO AHEAD AND FIND A INVERSE USING A CALCULATOR,
SO THE FIRST STEP IS TO ENTER MATRIX A,
AND WE'LL ALSO ENTER MATRIX B.
SO I'LL PRESS 2ND MATRIX, RIGHT ARROW TWICE FOR EDIT,
PRESS ENTER, ENTER THE DIMENSIONS, 3, ENTER, 3 ENTER,
AND ENTER IN ALL OF THE ELEMENTS.
IT'S IMPORTANT TO DOUBLE-CHECK
THIS BECAUSE IF ONE ELEMENT IS INCORRECT,
OF COURSE, WE'LL END UP GETTING THE WRONG ANSWER.
NOW, LET'S GO AHEAD AND ENTER MATRIX B,
SO I'LL PRESS 2ND MATRIX, RIGHT ARROW TO EDIT,
ARROW DOWN TO MATRIX B, ENTER, DIMENSIONS ARE 3x1,
ENTER THE ELEMENTS,
AND NOW WE'LL FIND THE INVERSE OF MATRIX A,
SO WE'LL GO BACK TO THE HOME SCREEN,
2ND MODE FOR QUIT,
AND THEN 2ND MATRIX, SELECT MATRIX A.
THE INVERSE KEY IS X TO THE -1, PRESS ENTER.
WE WANT TO CONVERT THESE DECIMALS TO FRACTIONS,
SO WE'RE GOING TO PRESS MATH, ENTER, ENTER.
NOTICE HOW WE COULD ONLY SEE THE FIRST TWO COLUMNS,
BUT IF WE ARROW TO THE RIGHT WE CAN SEE THAT THIRD COLUMN.
SO WE'LL GO AHEAD AND COPY THIS DOWN BY COLUMN.
FIRST COLUMN IS -1/16, 3/8, - 5/16.
2ND COLUMN IS 3/16, -1/8, AND -1/16.
AND THE THIRD COLUMN WILL BE 11/16, -1/8, AND 7/16.
AND OF COURSE, WE ALSO HAVE A INVERSE
ON THE RIGHT SIDE OF THE EQUATION.
NOW LOOKING AT THE LEFT SIDE OF THE EQUATION,
WE SHOULD RECOGNIZE THAT A INVERSE x A
IS GOING TO BE EQUAL TO A 3 x 3 IDENTITY MATRIX,
AND THEN THE IDENTITY MATRIX x MATRIX X WOULD JUST BE MATRIX X.
SO WE JUST HAVE X, Y, Z HERE.
AND THEN ON THE RIGHT SIDE WE HAVE A INVERSE x B.
LET'S FIRST MAKE SURE THIS PRODUCT IS DEFINED.
MATRIX A INVERSE IS A 3x3 MATRIX.
MATRIX B IS A 3x1 MATRIX,
SO BECAUSE THE NUMBER OF COLUMNS IN THE FIRST MATRIX
EQUALS THE NUMBER OF ROWS IN THE SECOND MATRIX
THIS PRODUCT IS DEFINED,
AND THE PRODUCT WILL BE A 3x1 MATRIX.
NOW AGAIN, WE'RE NOT GOING TO PERFORM
THIS MULTIPLICATION BY HAND,
BUT AGAIN, I WILL PUT A LINK ON THE SCREEN
TO A VIDEO THAT WILL SHOW HOW TO MULTIPLY MATRICES BY HAND.
FOR THIS VIDEO WE'RE GOING TO GO AHEAD AND FIND THIS PRODUCT
IN THE CALCULATOR.
WE'VE ALREADY ENTERED MATRIX A AND MATRIX B
INTO THE CALCULATOR,
SO WE JUST NEED TO FIND THE PRODUCT OF A INVERSE AND B.
SO WE'LL PRESS 2ND MATRIX,
ENTER, X TO THE -1 x 2ND, MATRIX, MATRIX B,
SO NUMBER 2, ENTER, AND ENTER.
NOTICE THE 3x1 MATRIX HAS THE ELEMENTS -1, 3, -2
WHICH MEANS OUR SOLUTION IS X = -1, Y = 3, Z = -2.
AS FAR AS AN ORDERED TRIPLE, WE HAVE -1, 3, -2.
OKAY, HOPE YOU FOUND THIS EXPLANATION HELPFUL.
THANK YOU FOR WATCHING.