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Chapter 9: Matrices and Determinants Study Guide

Section 9.1
Hints:

1.   A rectangular array of numbers arranged in rows and columns and placed in brackets is called a matrix.  The numbers inside the matrix are called elements.  When you give the dimensions of a matrix, you are giving the number of its rows and columns.  The number of rows is always listed first.

            Ex.: 

The matrix in the example has 2 rows and 3 columns.  You would say it is a matrix.

2.  An augmented matrix can be used to write a system of equations.  You may think of it as a kind of shorthand notation.  In an augmented matrix a vertical bar separates the columns of the matrix into two groups.  The coefficients of the system are placed to the left of the bar, and the constants are placed to the right of the bar.

            Ex.: 

3.  A matrix with 1s down the diagonal from upper left to lower right and 0s below the 1s is said to be in row-echelon form.  If you can change a system's augmented matrix into row-echelon form, then you will have the solution to the system.  Below is an example of a row-echelon matrix.

            Ex.: 

In order to turn an augmented matrix into a row-echelon form, you should use row operations.  Row operations are given on page 771.  

4.  The process of using row operations to solve linear systems is called Gaussian elimination.  The steps for using Gaussian elimination are on page 773.  Remember the numbers to the left of the bar are the coefficients of the variables!  The final steps after getting the augmented matrix in row-echelon form are to use back substitution to solve for the remaining variables. For example, the row-echelon matrix on the right below can be converted into the system given on the right and then solved by back substitution.

        Ex.:                               

5.   The method called Gauss-Jordan elimination just carries Gaussian Eeimination a little farther.  When you use Gauss-Jordan elimination, you use row operations to obtain a matrix to the left of the bar with 1s down the diagonal from upper left to lower right and 0s in every position above and below each 0.  When you have done this, the solution to the system will be the column to the right of the bar.  

            Ex.:    The solution to the system given by this matrix is (-1, 3, 2).

The steps for Gauss-Jordan elimination are on page 778.

Additional Exercises:
1.	Q:  Solve the system using Gaussian elimination.       Click Here for Answer
2.	Q:  Solve the system using Gauss-Jordan elimination.    Click Here for Answer

Section 9.2
Hints:

1.  Linear systems can have one solution, no solution, or infinitely many solutions.  Gaussian elimination can be used to determine the number of solutions for systems with three or more equations.    

2.  When using Gaussian elimination, if the bottom row to the left of the line becomes all 0s and the bottom row to the right of the line is a number that is NOT zero, then the system has no solution.

                Ex.: 

3.   When using Gaussian elimination, if the bottom row to the left and right of the line becomes all 0s, then the system has infinitely many solutions.

                Ex.:    

In a system with three equations and three unknowns and infinitely many solutions, it is possible to find an algebraic expression for x and y in terms of z.  In order to do this, you should first drop the bottom row of zeros;  then, take the remaining two rows and convert them back into a system.  This system can then be solved for x and y in terms of z.

Additional Exercises:
1.	Q:  Use Gaussian elimination to find the complete solution to the system of equations, or show that none exists.       Click Here for Answer
2.	Q:  Use Gaussian elimination to find the complete solution to the system of equations, or show that none exists.       Click Here for Answer

 

Section 9.3
Hints:

1.  The order of a matrix refers to its number of rows and columns.  When giving the order always list the number of rows first.  Each element in a matrix can be identified by the row and column it occupies.  For example, element is in row 3 and column two of the matrix.

 

 

 

 

2.  Two matrices are equal if and only if they have the same order and corresponding elements are equal.

 

 

        

3.   When adding or subtracting matrices simply add or subtract corresponding elements.  You cannot add or subtract two matrices of different orders.

 

4.  Properties of matrix addition are similar to the properties of with adding real numbers.  These properties are given on page 795.  

 

5.  Any matrix can be multiplied by a real number.  These real numbers are called scalars.  When multiplying by a scalar, you multiply every element in the matrix by the scalar.

 

6.  When multiplying matrices, you do not multiply corresponding elements.  Instead, you multiply rows by columns.  For the product of two matrices to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix.  The order of the answer matrix is the number of rows in the first matrix by the number of columns in the second matrix.                        

Ex.  Order--First Matrix          Order--Second Matrix                         Order--Answer Matrix

                2 x 3                                3 x 4                                   2 (rows from 1st) x 4 (columns from 2nd)

Notice that the columns in the first matrix match the rows in the second matrix.  Since this is the case, these two matrices can be multiplied together. There are 2 rows in the first matrix and 4 columns in the second matrix; therefore, the answer matrix will be 2 x 4.

 

7.  Remember!  Matrix multiplication is not commutative.  AB is not necessarily the same thing as BA.

 

 

Additional Exercises:
1.	Q:  Find 3A - 4B.       Click Here for Answer
2.	Q:  Find AB.       Click Here for Answer
3.	Q:   Find A(B - C).       Click Here for Answer

 

 

Section 9.4
Hints:

1.   A n x n square matrix whose main diagonal elements are 1s, while all other elements are 0s, is called the multiplicative identity matrix of order n.  The symbol used to designate this multiplicative identity matrix is  .

                       Ex.: 

2.  Some matrices have a multiplicative inverse.  When you multiply a matrix by its multiplicative inverse, you get an identity matrix, .  This is similar to getting a 1 when you multiply a real number by its reciprocal.  The symbol used for the multiplicative inverse of matrix A is .  A nonsquare matrix cannot have a multiplicative inverse.

3.  The process for finding the multiplicative inverse of a 2 x 2 matrix is

                        If , then .  

The matrix is invertible if and only if ad - bc does not equal zero.  If you look at the process carefully, you will see that the first factor requires you to multiply along the diagonals of the matrix and subtract.  To find the second factor, which is a matrix, reverse a and d and then negate b and c.

4.  The procedure for finding the multiplicative inverse of any invertible matrix is on page 813.    This procedure requires you to form an augmented matrix and then perform row operations.

5.  You may solve a system using a multiplicative inverse.  If , where A is the coefficient matrix and B is the constant matrix, has a unique solution, .  To solve the linear system of equations, you multiply and B to find X.

 

 

Additional Exercises:
1.	Q:  Find .              Click Here for Answer
2.	Q:  Find .              Click Here for Answer
3.	Q:  Write the linear system as a matrix equation and then solve the system using the inverse of the coefficient matrix.   Click Here for Answer

 

Section 9.5
Hints:

1.  Associated with every square matrix is a number called a determinant.  The determinant for a 

2 x 2 matrix  is found by computing

 

 

                                 .

 

This is the same process that you use to find the denominator in the first factor of the inverse of a 2 x 2 matrix.  You did this in 9.4.  It is not a new procedure.  Two vertical bars symbolize the determinant of a matrix.  If A represents the matrix, represents its determinant.

2.  Determinants can be used to solve a system of equations.  This method is called Cramer's Rule.  When using Cramer's Rule to solve a system with two equations and 2 unknowns, (x, y), you must find three different determinants.  One determinant is used as the denominator of both x and y.   One is used as the numerator of x, and one is used as the numerator of y.  If  , then

 

 

                    

 

 

Notice that when you solve for x, the column that is the coefficients of x (the a's) is replaced by column of  constants (the c's), and when you solve for y, the column that is the coefficients of y (the b's) is replaced by column of  constants (the c's).  Think of it this way.  When you look for x, you replace the x coefficients, and when you look for y you replace the y coefficients.  The denominator is always the determinant of the coefficients of x and y.

 

 

3.  There is more work involved in finding the determinant of a 3 x 3 matrix.  The definition in the middle of page 826 is the one that you will want to use.  At the bottom of page 826 are some helpful hints for evaluating the determinant of a 3 x 3 matrix.  These are steps that you need to memorize.

 

                   

 

 

Each 2 x 2 determinant in the formula is called a minor of an element.  It is what remains after deleting the row and column of that element.  Using this method to find the determinant of a 3 x 3 is called expansion by minors. 

 

 

 

4. The formula given above is expansion of the first column (the a's).  However, you can expand by minors about any row or column. When a row or column contains zeros, consider using it for your expansion.  That will simplify the arithmetic.

 

 

 

5. A system with three equations and three unknowns may be solved using Cramer's Rule.  The step require you to find four third-order determinants.  The process is similar to solving a system with only two variables.

 

If  , then .

      

 

 

 

 

 

 

Each third order determinant is found by expansion by minors.

 

6.  When using Cramer's Rule, if D = 0 and at least one of the determinants in the numerator is not 0, then the system is inconsistent.  The solution is the empty set.  If D = 0 and all the determinants in the numerators are 0, then the equations in the system are dependent.  The system has infinitely many solutions. This hold true regardless of the size of the system.

 

 

 

Additional Exercises:
1.	Q:  Find the determinant of A.       Click Here for Answer
2.	Q:  Find the determinant of A.       Click Here for Answer
3.	Q:  Solve the system using Cramer's Rule.       Click Here for Answer

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