Lecture 3 Matrices and Linear Equations AMath 352 Fri Apr 2 1 17 Use eq 1 to eliminate x in eq 2 Subtract 3 times eq 1 from eq 2 3x y 3 x 2y 2 3 3 7y 7 y 1 Now substitute y 1 into eq 1 and solve for x x 2 1 3 x 1 Now check 1 2 1 3 1 1 3 yes 2 yes Linear Equations Solve for x and y x 2y 3 3x y 2 2 17 Now substitute y 1 into eq 1 and solve for x x 2 1 3 x 1 Now check 1 2 1 3 1 1 3 yes 2 yes Linear Equations Solve for x and y x 2y 3 3x y 2 Use eq 1 to eliminate x in eq 2 Subtract 3 times eq 1 from eq 2 3x y 3 x 2y 2 3 3 7y 7 y 1 2 17 Now check 1 2 1 3 1 1 3 yes 2 yes Linear Equations Solve for x and y x 2y 3 3x y 2 Use eq 1 to eliminate x in eq 2 Subtract 3 times eq 1 from eq 2 3x y 3 x 2y 2 3 3 7y 7 y 1 Now substitute y 1 into eq 1 and solve for x x 2 1 3 x 1 2 17 Linear Equations Solve for x and y x 2y 3 3x y 2 Use eq 1 to eliminate x in eq 2 Subtract 3 times eq 1 from eq 2 3x y 3 x 2y 2 3 3 7y 7 y 1 Now substitute y 1 into eq 1 and solve for x x 2 1 3 x 1 Now check 1 2 1 3 1 1 3 yes 2 yes 2 17 Use eq 1 to eliminate x in eqs 2 and 3 Subtract 2 times the rst eq from the second add the rst eq to the third 2x 3y 4z 2 x 2y 3z 5 2 4 y 2z 3 x 6y z x 2y 3z 2 4 4y 4z 6 Linear Equations Cont This technique can be extended to any number of linear equations in the same number of unknowns Solve for x y and z x 2y 3z 4 2x 3y 4z 5 x 6y z 2 3 17 Linear Equations Cont This technique can be extended to any number of linear equations in the same number of unknowns Solve for x y and z x 2y 3z 4 2x 3y 4z 5 x 6y z 2 Use eq 1 to eliminate x in eqs 2 and 3 Subtract 2 times the rst eq from the second add the rst eq to the third 2x 3y 4z 2 x 2y 3z 5 2 4 y 2z 3 x 6y z x 2y 3z 2 4 4y 4z 6 3 17 Substitute this into the second eq to nd y 2 3 y 0 Substitute for y and z in the rst eq to nd x 2 0 3 4 x 1 2 3 2 3 2 Linear Equations Cont x 2y 3z 4 y 2z 3 4y 4z 6 Subtract 4 times the second eq from the third to nd 4y 4z 4 y 2z 6 4 3 12z 18 z 3 2 4 17 Substitute for y and z in the rst eq to nd x 2 0 3 4 x 3 2 1 2 Linear Equations Cont x 2y 3z 4 y 2z 3 4y 4z 6 Subtract 4 times the second eq from the third to nd 4y 4z 4 y 2z 6 4 3 12z 18 z 3 2 Substitute this into the second eq to nd y 2 3 y 0 3 2 4 17 Linear Equations Cont x 2y 3z 4 y 2z 3 4y 4z 6 Subtract 4 times the second eq from the third to nd 4y 4z 4 y 2z 6 4 3 12z 18 z 3 2 Substitute this into the second eq to nd y 2 3 y 0 Substitute for y and z in the rst eq to nd x 2 0 3 4 x 1 2 3 2 3 2 4 17 Linear Equations Cont Check x 1 2 y 0 z 3 2 x 2y 3z 4 2x 3y 4z 5 x 6y z 2 4 yes 2 4 5 yes 1 2 1 2 1 2 9 2 3 2 3 2 2 yes 5 17 Matrix Notation We don t have to write x y and z so many times x 2y 3z 4 2x 3y 4z 5 x 6y z 2 can be written as 1 2 2 3 3 4 1 6 1 x y z 4 5 2 The 3 by 3 array of numbers on the left is a matrix and the product of this matrix with the vector x y z T of unknowns is just the left hand side of the above equations 1 2 2 3 3 4 1 6 1 x y z x 2y 3z 2x 3y 4z x 6y z 6 17 Eliminate x from the second and third eqs by subtracting 2 times the rst eq from the second and adding the rst eq to the third Eliminate y from the third eq by subtracting 4 times the second eq 1 2 3 0 1 2 3 0 4 4 4 6 3 4 1 0 2 0 0 1 2 3 12 18 Solving Equations Using Matrix Notation Append the right hand side vector to the matrix 1 2 2 3 4 3 4 5 1 6 1 2 7 17 Eliminate y from the third eq by subtracting 4 times the second eq 3 4 1 0 2 0 0 1 2 3 12 18 Solving Equations Using Matrix Notation Append the right hand side vector to the matrix Eliminate x from the second and third eqs by subtracting 2 times the rst eq from the second and adding the rst eq to the third 1 2 2 3 4 3 4 5 1 6 1 2 2 3 1 4 0 1 2 3 6 0 4 4 7 17 Solving Equations Using Matrix Notation Append the right hand side vector to the matrix Eliminate x from the second and third eqs by subtracting 2 times the rst eq from the second and adding the rst eq to the third Eliminate y from the third eq by subtracting 4 times the second eq 1 2 2 3 4 3 4 5 1 6 1 2 2 3 1 4 0 1 2 3 6 0 4 4 2 3 1 4 0 1 2 3 18 0 12 0 7 17 Therefore z y 2 3 y 0 18 12 3 2 3 2 x 3 4 x 3 2 1 2 Backsolve Recall that this means 2 3 1 4 0 1 2 3 18 0 12 0 x 2y 3z 4 y 2z 3 12z 18 8 17 Backsolve Recall that this means 2 3 1 4 0 1 2 3 …