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Linear Functions and Matrices

Linear Functions and Matrices. Linear literal equations. A literal equation in x is an equation whose solution will be expressed in terms of pronumerals rather than numbers. Example: Solve the following for x . a. px − q = r b. ax + b = cx + d Solution:.

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Linear Functions and Matrices

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  1. Linear Functions and Matrices Linear literal equations A literal equation inxis an equation whose solution will be expressed in terms of pronumerals rather than numbers. Example:Solve the following forx. a.px− q = rb.ax + b = cx + d Solution:

  2. Simultaneous literal equations Simultaneous literal equations are solved by the methods of solution of simultaneousequations, i.e. substitution and elimination. Example:Solve the following pairs of simultaneous equations forx. ay = ax + c y = bx + d bax − y = c x + by = d Solution:

  3. Product of gradients of two perpendicular straight lines If two straight lines are perpendicular to each other, the product of their gradients is −1. m1m2 =−1 Distance between two points Midpoint The midpoint of a straight line joining (x1,y1)and(x2,y2) is the point.

  4. The angle α between intersecting straight lines α = θ2 - θ1 Example:A fruit and vegetable wholesaler sells 30kg of hydroponic tomatoes for $148.50 and 55 kg ofhydroponic tomatoes for $247.50. Find a linear model for the cost,C dollars, to buy x kg ofhydroponic tomatoes. How much would 20 kg of tomatoes cost? Solution: Let(x1,C1) =(30,148.5) and (x2,C2)=(55, 247.5). The equation of the straight line is given by solving the following simultaneous equation : 148.5 = m30 + c 1 247.5 = m 55 + c 2 By subtracting 2-1 we get: 99 = 25 m m = 3.96

  5. By substituting m= 3.96 in 1 we get: 148.5 =3.96x30 + c c = 29.7 Therefore the straight line is given by the equation C =3.96x+29.7 Substitute x =20:C = 3.96 × 20 + 29.7=108.9 It would cost $108.90 to buy 20 kg of tomatoes. Applications in linear functions Example:There are two possible methods for paying gas bills: method A: a fixed charge of $25 per quarter + 50c per unit of gas used method B: a fixed charge of $50 per quarter + 25c per unit of gas used. Determine the number of units which must be used before method B becomes cheaper thanmethod A. Solution: Let C1 = charge in $ using method A C2 = charge in $ using method B x = number of units of gas used. Now C1 = 25 + 0.5x C2 = 50 + 0.25x C1 = C2 when A and B are equal 25 + 0.5x = 50 + 0.25x 0.25x = 25 x =100

  6. Review of matrix arithmetic The size ordimensionof a matrix is described by specifying the number of rows (m) and the number of columns (n). The dimension is writtenm × n. The dimensions of thefollowing matrices in order are:3 × 2, 1 × 4, 3 × 3, 1 × 1 Addition will be defined for two matrices only when they have the same dimension. The sum is found by adding corresponding elements.Subtraction is defined in a similar way. IfAis anm × nmatrix andkis a real number,kAis defined to be anmnmatrix whoseelements arektimes the corresponding element ofA.

  7. The productABis defined only if the number of columns ofAis the same as thenumber of rows ofB.ABis a 2 × 2 matrix. The identity matrix for the family ofn × nmatrices is the matrix with ones in the ‘top left’ to ‘bottom right’ diagonal and zeros in all other positions. This is denoted byI. IfAandBare square matrices of the same dimension andAB = BA = IthenAis said tobe the inverse ofBandBis said to be the inverse ofA. The inverse ofAis denoted byA−1. det (A) = a d − bcis thedeterminantof matrixA. A square matrix is said to beregularif its inverse exists. Those square matrices which donot have an inverse are calledsingularmatrices. The determinant of a singular matrixis 0.

  8. Using the ti- enspire for performing different operations with matrices The matrix is obtained by pressing ctrl x and the arrow by pressing ctrl sto

  9. Example: Solution:a det = 3x6 - 1x2 = 16

  10. Solving systems of linear simultaneous equations in two variables Inverse matrices can be used to solve certain sets of simultaneous linear equations. Considerthe equations: 3x − 2y = 5 5x − 3y = 9 This can be written as: The determinant ofAis 3( −3) − 5(−2) = 1 which is not zero and soA−1 exists. Multiplying the matrix equation on both sides on the left-hand sidebyA−1and using the fact thatA−1A = Iyields the following:

  11. Example:Solve the simultaneous equations:3x − 2y = 6 7x + 4y = 7 Solution: Using the Ti- enspire

  12. Example:The simultaneous equations 2x+ 3y = 6 and 4x + 6y = 12 have infinitely many solutions.Describe these solutions through the use of a parameter. Solution:Let y =ℷ then x = -3(ℷ - 2) 2 Using the ti-enspire we replace ℷ with c1 Example:Consider the simultaneous linear equations(m − 2)x + y = 2 andmx + 2y = k Find the values ofmandksuch that the system of equations has: aa unique solutionbno solutioncinfinitely many solutions Solution Using a CAS calculator to find the solution.

  13. aThe solution is unique ifm = 4 andkcan be any real number. bIfm = 4, the equations become 2x + y = 2 and 4x + 2y = k There is no solution ifm = 4 andk ≠ 4. cIfm = 4 andk = 4 there are infinitely many solutions as the equations are thesame.

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