Lecture 3: Matrices and Linear EquationsAMath 352Fri., Apr. 21 / 17Linear EquationsSolve for x and y:x + 2y = 33x − 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 yes3 · 1 − 1?= 2 yes2 / 17Linear EquationsSolve for x and y:x + 2y = 33x − 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 yes3 · 1 − 1?= 2 yes2 / 17Linear EquationsSolve for x and y:x + 2y = 33x − 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 yes3 · 1 − 1?= 2 yes2 / 17Linear EquationsSolve for x and y:x + 2y = 33x − 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 yes3 · 1 − 1?= 2 yes2 / 17Linear Equations, Cont.This technique can be extended to any number of linear equationsin the same number of unknowns. Solve for x, y, and z:x + 2y + 3z = 42x + 3y + 4z = 5−x − 6y + z = 2.Use eq. 1 to eliminate x in eqs. 2 and 3. Subtract 2 times the first eq.from the second; add the first 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 / 17Linear Equations, Cont.This technique can be extended to any number of linear equationsin the same number of unknowns. Solve for x, y, and z:x + 2y + 3z = 42x + 3y + 4z = 5−x − 6y + z = 2.Use eq. 1 to eliminate x in eqs. 2 and 3. Subtract 2 times the first eq.from the second; add the first 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 / 17Linear Equations, Cont.x + 2y + 3z = 4−y − 2z = −3−4y + 4z = 6.Subtract 4 times the second eq. from the third to find:−4y + 4z − 4(−y − 2z) = 6 − 4 · (−3) ⇒ 12z = 18 ⇒ z =32.Substitute this into the second eq. to find:−y − 2 ·32= −3 ⇒ y = 0.Substitute for y and z in the first eq. to find:x + 2 · 0 + 3 ·32= 4 ⇒ x = −12.4 / 17Linear Equations, Cont.x + 2y + 3z = 4−y − 2z = −3−4y + 4z = 6.Subtract 4 times the second eq. from the third to find:−4y + 4z − 4(−y − 2z) = 6 − 4 · (−3) ⇒ 12z = 18 ⇒ z =32.Substitute this into the second eq. to find:−y − 2 ·32= −3 ⇒ y = 0.Substitute for y and z in the first eq. to find:x + 2 · 0 + 3 ·32= 4 ⇒ x = −12.4 / 17Linear Equations, Cont.x + 2y + 3z = 4−y − 2z = −3−4y + 4z = 6.Subtract 4 times the second eq. from the third to find:−4y + 4z − 4(−y − 2z) = 6 − 4 · (−3) ⇒ 12z = 18 ⇒ z =32.Substitute this into the second eq. to find:−y − 2 ·32= −3 ⇒ y = 0.Substitute for y and z in the first eq. to find:x + 2 · 0 + 3 ·32= 4 ⇒ x = −12.4 / 17Linear Equations, Cont.x + 2y + 3z = 42x + 3y + 4z = 5−x − 6y + z = 2.Check: x = −12, y = 0, z =32.−12+92?= 4 yes2 · (−12) + 4 ·32?= 5 yes12+32?= 2 yes.5 / 17Matrix NotationWe don’t have to write x, y, and z so many times!x + 2y + 3z = 42x + 3y + 4z = 5−x − 6y + z = 2.can be written as1 2 32 3 4−1 −6 1xyz=452.The 3 by 3 array of numbers on the left is a matrix, and theproduct of this matrix with the vector [x, y, z]Tof unknowns isjust the left-hand side of the above equations:1 2 32 3 4−1 −6 1xyz=x + 2y + 3z2x + 3y + 4z−x − 6y + z.6 / 17Solving Equations Using Matrix NotationAppend the right-hand side vector to the matrix:1 2 3 | 42 3 4 | 5−1 −6 1 | 2.Eliminate x from the second and third eqs. by subtracting 2 timesthe first eq. from the second and adding the first eq. to the third:1 2 3 | 40 −1 −2 | −30 −4 4 | 6.Eliminate y from the third eq. by subtracting 4 times the secondeq.:1 2 3 | 40 −1 −2 | −30 0 12 | 18.7 / 17Solving Equations Using Matrix NotationAppend the right-hand side vector to the matrix:1 2 3 | 42 3 4 | 5−1 −6 1 | 2.Eliminate x from the second and third eqs. by subtracting 2 timesthe first eq. from the second and adding the first eq. to the third:1 2 3 | 40 −1 −2 | −30 −4 4 | 6.Eliminate y from the third eq. by subtracting 4 times the secondeq.:1 2 3 | 40 −1 −2 | −30 0 12 | 18.7 / 17Solving Equations Using Matrix NotationAppend the right-hand side vector to the matrix:1 2 3 | 42 3 4 | 5−1 −6 1 | 2.Eliminate x from the second and third eqs. by subtracting 2 timesthe first eq. from the second and adding the first eq. to the third:1 2 3 | 40 −1 −2 | −30 −4 4 | 6.Eliminate y from the third eq. by subtracting 4 times the secondeq.:1 2 3 | 40 −1 −2 | …