Lesson OVERVIEW of LESSONS on determinants of 2x2-matrices and Cramer's rule for systems in 2 unknowns

Algebra ->  Matrices-and-determiminant -> Lesson OVERVIEW of LESSONS on determinants of 2x2-matrices and Cramer's rule for systems in 2 unknowns      Log On


   

This Lesson (OVERVIEW of LESSONS on determinants of 2x2-matrices and Cramer's rule for systems in 2 unknowns) was created by by ikleyn(52756) About Me : View Source, Show
About ikleyn:

OVERVIEW of LESSONS on determinants of 2x2-matrices and Cramer's rule for systems in 2 unknowns


For your convenience this file contains the list of my lessons on determinants of 2x2 matrices and the Cramer's rule for systems in 2 unknowns in the logical order.
Then the lessons are listed again with short annotations of their contents.

The list of lessons

    - What is a matrix?,
    - Determinant of a 2x2-matrix,
    - HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule),
    - Solving systems of linear equations in two unknowns using the Cramer's rule,
    - Solving word problems by the Cramer's rule after reducing to systems of linear equations in two unknowns,
    - Solving systems of non-linear equations in two unknowns using the Cramer's rule  and
    - Determinant of a 2x2-matrix and the area of a parallelogram and a triangle.

The lessons with short annotations

Lesson  Determinant of a 2x2-matrix

    The  determinant  of a  2x2-matrix  A  = %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29  is the number   d = a%2Ad+-+b%2Ac.

    The determinant of a matrix   %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29   is denoted as   |matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29|  or as  det %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29,      
    so   |matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29| = det %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29 = ad+-+bc.

    The pattern to calculate the determinant is shown in  Figure 1.

    The determinant of a  2x2-matrix is the product of its upper left

                

    Figure 1. The pattern to calculate
   the determinant of a 2x2 matrix
    and lower right elements  (at the diagonal shown in black)  minus the product of the lower left and upper right elements  (at the diagonal shown in red).

    Examples.  Calculate determinants   |matrix%282%2C2%2C+1%2C+0%2C+0%2C+1%29|;   |matrix%282%2C2%2C+1%2C+2%2C+0%2C+1%29|;   |matrix%282%2C2%2C+0%2C+0%2C+0%2C+0%29|;   |matrix%282%2C2%2C+1%2C+2%2C+0%2C+0%29|;   |matrix%282%2C2%2C+1%2C+2%2C+2%2C+1%29|;   |matrix%282%2C2%2C+1%2C+2%2C+-2%2C+1%29%29|;   |matrix%282%2C2%2C+1%2C+2%2C+2%2C+4%29|;   |matrix%282%2C2%2C+1%2C+2%2C+-2%2C+-4%29|.


Lesson  HOW TO solve system of linear equations in two unknowns using determinant (Cramer's rule)

    If
        

    is a system of two linear equations in two unknown  x  and  y,  where  a%5B11%5D, a%5B12%5D, a%5B21%5D, a%5B22%5D  are coefficients and  b%5B1%5D, b%5B2%5D  are right side constants,
    then the solution of this system is  (the Cramer's rule)

        x = = ,      y = =

    Here the matrix  %28matrix%282%2C2%2C+a%5B11%5D%2C+a%5B12%5D%2C+a%5B21%5D%2C+a%5B22%5D%29%29  in the denominators is the coefficient matrix of the system formed by its coefficients.  It is assumed that
    the coefficient matrix has the non-zero determinant.  This is the necessary condition for the Cramer's rule applicability.

    The matrix in the numerator for the  first unknown  x  is obtained from the coefficient matrix after replacing its  first column  by the right sides vector  %28matrix%282%2C+1%2C+b%5B1%5D%2C+b%5B2%5D%29%29.
    The matrix in the numerator for the  second unknown  y  is obtained from the coefficient matrix after replacing its  second column  by the right sides vector  %28matrix%282%2C1%2C+b%5B1%5D%2C+b%5B2%5D%29%29.

    Example 1.  Solve the system of linear equations in two unknowns

            system+%282x+%2B+y+=+5%2C%0D%0A4x+%2B+6y+=++14%29

    Example 2.  Solve the system of linear equations in two unknowns

            system+%282x+%2B+y+=+5%2C%0D%0A-4x+%2B+6y+=++-2%29


Lesson  Solving systems of linear equations in two unknowns using the Cramer's rule

    Example 1.  Solve the system of linear equations in two unknowns

            system+%284x+%2B+y+=+17%2C%0D%0A3x+-+y+=++4%29

    Example 2.  Solve the system of linear equations in two unknowns

            system+%284u+-3v+=+2%2C%0D%0A5u+%2B4v+=+3%29

    Example 3.  Solve the system of linear equations in two unknowns

            system+%282x+%2B+3y+=+2%2C%0D%0A4x+%2B+6v+=+4%29

    Example 4.  Solve the system of non-linear equations in two unknowns

            system%282%2Fx+%2B+3%2Fy+=+4%2C+%0D%0A4%2Fx+%2B+1%2Fy+=+3%29


Lesson  Solving word problems by the Cramer's rule after reducing to systems of linear equations in two unknowns

    Problem 1.  A total  820  tickets were sold for a game for a total of  $9,128.
                       If adults tickets were sold for  $12.00  and children tickets were sold for  $8.00,  how many of each kind of tickets were sold?

    Problem 2.  Mr. Sullivan had  $20,000  to invest.  He invested part at  3%  and the rest at  2%.  After one year he earned $520.
                       How much did he invest at each rate?

    Problem 3.  One acid solution contains  30%  of acid.  Another acid solution contains  48%  of the same acid.
                       How many liters of each solution should be mixed to produce  36  liters of a solution which is  40%  of acid?

    Problem 4.  It takes a boat  3  hours to travel  24  miles downstream a river and  4  hours to return back.
                       What is the speed of the boat in still water?  What is the speed of the current of the river?

    Problem 5.  An airplane covers a distance of  1800  miles in  4  hours flying with the wind.  The return trip against the same wind takes  4.5  hours.
                       What is the speed of the airplane in still air?  What is the speed of the wind?


Lesson  Solving systems of non-linear equations in two unknowns using the Cramer's rule

    Problem 1.  Solve the system of non-linear equations in two unknowns

            system+%282%2Fx+%2B+1%2Fy+=+11%2C%0D%0A3%2Fx+-+5%2Fy+=+10%29.

    Problem 2.  Solve the system of non-linear equations in two unknowns

            .

    Problem 3.  Solve the system of non-linear equations in two unknowns

            .

    Problem 4.  Solve the following system of exponential equation for  x  and  y:

            .


Lesson  Determinant of a 2x2-matrix and the area of a parallelogram and a triangle

There is a connection between the determinants of 2x2-matrices and areas of parallelograms:

Let  A = %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29  be a  2x2-matrix.  Then the modulus  (the absolute value)  of the determinant of the matrix  A,  |det %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29|,  is equal to the area of the parallelogram
which is built in a coordinate plane on vectors  u = %28matrix%282%2C1%2C+a%2C+c%29%29  and  v = %28matrix%282%2C1%2C+b%2C+d%29%29  that are the columns of the matrix  A.

Conversely,  if  u = %28matrix%282%2C1%2C+a%2C+c%29%29  and  v = %28matrix%282%2C1%2C+b%2C+d%29%29  are vectors in a coordinate plane,  then the area of the parallelogram which is built on these vectors as on sides is equal to the modulus of the determinant,  |det %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29|,  of the  2x2-matrix  A = %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29  whose columns are the given vectors.

    Example 1.  Find the area of a parallelogram in a coordinate plane with one vertex at the origin if its sides released from this vertex are the vectors with coordinates
                        u = (2,1)  and  v = (1,2).

    Example 2.  Find the area of a parallelogram in a coordinate plane with one vertex at the origin if its sides released from this vertex are the vectors with coordinates
                        u = (1,3)  and  v = (3,1).


There is similar connection between the determinants of 2x2-matrices and areas of triangles:

Let  A = %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29  be a  2x2-matrix.  Then the modulus  (the absolute value)  of the determinant of the matrix  A,  |det %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29|,  is equal to the doubled area of the triangle
which is built in a coordinate plane on vectors  u = %28matrix%282%2C1%2C+a%2C+c%29%29  and  v = %28matrix%282%2C1%2C+b%2C+d%29%29  that are the columns of the matrix  A:  |det %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29| = 2S%5BDELTA%5D.

Conversely,  if  u = %28matrix%282%2C1%2C+a%2C+c%29%29  and  v = %28matrix%282%2C1%2C+b%2C+d%29%29  are vectors in a coordinate plane,  then the area of the triangle which is built on these vectors as on sides is equal to the half of the modulus of the determinant,  |det %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29|,  of the  2x2-matrix  A = %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29  whose columns are the given vectors:  S%5BDELTA%5D = 1%2F2.|det %28matrix%282%2C2%2C+a%2C+b%2C+c%2C+d%29%29|.

    Example 3.  Find the area of a triangle in a coordinate plane with one vertex at the origin if its sides released from this vertex are the vectors with coordinates
                        u = (2,1)  and  v = (1,2).

    Example 4.  Find the area of a triangle in a coordinate plane with one vertex at the origin if its sides released from this vertex are the vectors with coordinates
                        u = (1,3)  and  v = (3,1).


Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.

This lesson has been accessed 2759 times.