SCIE 4101, Spring 011 Miller Math Review Packet #4 Algebra II (Part 1) Notes Matrices A matrix is a rectangular arra of numbers. The order of a matrix refers to the number of rows and columns the matrix has: a 3 (read b 3 ) matrix has rows of 3 numbers in each row; a 4 1 matrix has 4 rows of 1 number in each. Two matrices of the same order can be added or subtracted b adding or subtracting the corresponding elements. For example, 5 3 5 1 7 7 6 10 + = 1 4 3 4 6 6 An matrix can be multiplied b a number (called a scalar) b multipling each element of the matrix b that scalar. 1 1 3 3 Ex. 3 7 = 1 6 Multiplication of matrices can be somewhat complicated. First of all, unlike standard multiplication, it is not commutative-- A B B A. Secondl, the number of columns in the left matrix must equal the number of rows in the right matrix. So, for example, 4 1 4 1 1 1 works (3 columns = 3 rows), but 1 1 would not. 1 1 5 1 1 5 To multipl matrices, ou multipl the elements in the first row of the left matrix b the corresponding elements in the first column of the right matrix: 4 1 1 1 1 3 1 1 7 i + i + i = = 1 1 5 Now, multipl the first row of the left matrix b the corresponding elements of the second column of the right matrix: 4 1 1 1 7 1 4 0 1 1 7 5 i + i + i = = 1 1 5 When ou finish the first row, repeat the procedure using the second row, and so on: 4 1 1 1 7 5 10 = 3 4 85 1 1 5
1 0 0... 0 1 0... The multiplicative identit for an n n matrix is. 0 0 1............... Multipling a square matrix b the identit matrix of the same order leaves the matrix unchanged. If the product of two square matrices is the identit matrix, then the matrices are multiplicative inverses of each other. 7 3 5 5 1 0 Example: 8 7 = 8 3 0 1 5 5 The determinant of a matrix is a special number associated with an square matrix. If the determinant of a matrix is not equal to 0, the matrix has an inverse. To find the determinant of a matrix, multipl the numbers as follows: det a b a b ad bc c d = = c d 5 6 5 6 det 5 3 1 6 1 1 3 = = = 1 3 Example: ( )( ) ( )( ) Calculating the determinant for larger matrices gets a little trickier. The simplest wa to find it manuall is to extend the matrix and then multipl along the diagonals, summing one wa and subtracting the other: a b c a b c a b det d e f = d e f d e = aei + bfg + cdh gec hfa idb g h i g h i g h Example: 8 1 8 1 8 det 3 1 0 = 3 1 0 3 1 4 7 5 4 7 5 4 7 = ()( 1)(5) + (8)(0)(4) + ( 1)( 3)(7) (4)( 1)( 1) (7)(0)() (5)( 3)(8) = 10+ 0+ 1 4 0+ 10= 17 There are three was to use matrices to solve sstems of linear equations. For all three methods, the equations must be written in standard form ( Ax+ B+... = C ). Method #1 (requires a TI-83+ or greater calculator): If there are n equations (and n variables), create a matrix that is n n+1 (i.e. for a 3-equation sstem, create a 3 4 matrix). Find the RREF() of that matrix. The last column of the result are the answers.
5x 4 6z= 1 Example: Solve x + 3 + 4z = 15 3x 7 5z = 15 Step 1 >~~ÍÂÍ Í Step Step 3 Enter the coefficients of the variables, and the constants into the matrix: 5 4 6 1 3 4 15 3 7 5 15 5>~ƒŒ Step 4 >ÍÍ The solution is ( 1,1, 5). That is x= 1, = 1, and z= 5. Method #: If ou do not have a calculator, ou can set up the sstem as follows (using the same example): 5 4 6 x 1 x 3 4 15 =, and solve for b finding the inverse of the first matrix and 3 7 5 z 15 z multipling it b the constants matrix. This sort of manipulation is reall beond the scope of this course.
ax+ b= c Method #3 (Cramer s Rule): For a given sstem of linear equations, which can be dx+ e= f a b x c written in matrix form as d e = f, x and can be found b the following: a b D=, D d e x = c b f e, D = a d c f (calculate the determinant of the coefficients, then replace the column for each variable with the constants) D x x= and D D = D This also works for linear equations with three or more variables just swap the constants matrix values for the column containing that variable s coefficients. Using our same example, we set up the sstem as follows: 5 4 6 x 1 3 4 15 = 3 7 5 z 15 5 4 6 D= 3 4 3 7 5 The determinant of the denominator is 5 4 6 5 4 ( ) 3 4 3 =75 48 84 54 140 40 = 7 3 7 5 3 7 D x 1 4 6 1 4 = 15 3 4 15 3 =7 15 7 5 15 7 D 5 1 6 5 1 = 15 4 15= 7 3 15 5 3 15 D z 5 4 1 5 4 = 3 15 3 =135 3 7 15 3 7 Dx Dz Using Cramer s Rule, we can write x=, =, z=. D D D D x 7 D 7 x= = = 1 = = = 1 D 7 D 7 D D z 135 z= = = 5 D 7
Quadratic Functions As we know, a quadratic function is a function whose general equation is = ax + bx+ c and whose graph is a parabola. The lowest point (or highest, if a is negative) is called the vertex. It turns out we can also write this function as ( ) k= a x h where a, h, and k are constants, and the vertex is at the point ( h, k ). This is called the vertex form of the parabola. We can use the completing the square method to transform the function b from standard form to vertex form. (This can often be easier than calculating x= and then a substituting this back in to find.) Example: Rewrite the equation Solution: x x = 6 + = x 6x + = x x+ 9 6 9 x x = 6 + into vertex form and identif the vertex. + 7= ( x 3) The coordinates of the vertex are ( 3, 7) To write the equation for a quadratic function from its graph, if we know the vertex and a point, we can plug this into the vertex form and solve for a: 6 4 - -1 1 3 - -4-6 x ( 5) = a( x ) Plug in ( 3,1 ) + = a( ) = a( ) 1 5 3 6 1 a= 6 ( x ) + 5= 6 If we do not know the vertex, we will need three points to set up three equations to solve for a, b, and c.
Imaginar and Complex Numbers As we saw in our Algebra I review, if the discriminant of a quadratic equation is negative, we end up with imaginar roots to the equation. To make sense out of imaginar numbers, the first step is to define a number whose square is 1. This number is called i. More formall, i =1 i= 1 A number such as 36 can now be defined in terms of i, and it can be written this wa: ( )( ) 36 = 1 36 = 1 i 36 = ii 6= 6i From this, we can see that an imaginar number is the product of a real number and i. Now, let s look at a quadratic equation with imaginar roots: x 10x+ 34= 0 ( )( ) 10± 100 4 1 34 x= 10± 36 10± 6i x= = = 5± 3i A number such as 5+ 3i is called a complex number. The 5 is called the real part of 5 of 5+ 3i and the coefficient 3 is called the imaginar part of 5+ 3i. Complex numbers are written in the form a+ bi, where a and b are real numbers. Numbers such as 5+ 3i and 5 3i are called complex conjugates. You might notice, if ou recall our discussion of multipling and factoring binomials that these look ver like a difference of squares.