Chapter 9, (js), Systems of Equations and Inequalities

Gauss-Jordan Elimination (putting a matrix in reduced row-echelon form)

After putting a matrix in row-echelon form, obtain zeros above each leading entry by adding multiples of the row containing that entry to the rows above it. Begin with the last leading entry and work up.

What is the cofactor Aij of the element aij?

Aij = (-1)^i+j(Mij) The cofactor of aij is just the minor of aij multiplied by either- 1 or 1.

How do you find the inverse of an n by n Matrix?

Enlarge it to an n x 2n Matrix and put the identity Matrix on the right hand side. Use Elementary row operations to change the left side into the identity Matrix. When finished the right side has been transformed into the inverse of The Matrix.

What are the criteria for having a matrix in row-echelon form, and reduced row-echelon form?

For row-echelon form: - The first nonzero number in each row is 1. This is called the leading entry. - The leading entry in each row is to the right of the leading entry in the row immediately above it. - All rows consisting entirely of zeros are at the bottom of the matrix. For reduced row-echelon form: - Every number above and below each leading entry is zero.

As with operations performed on a linear system in several variables in Gaussian elimination, you can do what 3 things to an augmented matrix?

1. Add a multiple of one row to another. 2. Multiply a row by a nonzero constant. 3. Interchange two rows.

If you have a system of linear equations, what operations can you perform on it to get an equivalent triangular system (ie, Gaussian Elimination).

1. Add a nonzero multiple of one equation to another. 2. Multiply an equation by a nonzero constant. 3. Interchange the positions of 2 equations.

A systematic way to put a matrix in row-echelon form:

1. Obtain a 1 in the top left corner, then obtain zeros below that 1 by adding appropriate multiples of the first row to the rows below it. 2. Next, obtain a leading entry in the next row, and then obtain zeros below that 1. 3. At each stage make sure that every leading entry is to the right of the leading entry in the row above it - rearrange rows if necessary.

How do you know if a matrix has an inverse?

A matrix has an inversr if the determinant is not zero.

Describe the possible solutions of a system of linear equations in 2 variables.

A single point if the lines cross, no solution if the lines are parallel ("inconsistent"), infinitely many solutions if the lines coincide (dependent).

What is an augmented matrix?

A system of linear equations written as a matrix; only the coefficients and constants that appear in the equations appear in the matrix.

A linear system may have one solution, no solution, or infinitely many solutions. 1. No solution - if the row-echelon form has a row with 0 = c, where c isn't 0, the system has no solution; it's inconsistent. 2. One solution - each variable in the row-echelon form is a leading variable, then the system has one solution. 3. Infinitely many solutions - if the variables in the row-echelon form are not all leading variables, and if the system is not inconsistent, then it has infinitely many solutions.

Do example 7 on pg 671.

You can use matrices to solve a system of linear equations. Convert the system of equations into an equivalent matrix equation. (Coefficient matrix "A" × variable matrix "X" = constant matrix "B"). AX =B. Multiply both sides by A^-1. The left side becomes the Identity Matrix × X, so you have IX = A^-1(B), and then X = A^-1(B)

If A is a square n×n matrix that has an inverse A^-1, and if X is a variable matrix and B a known matrix, both with n rows, then the solution of the matrix equation AX = B is given by X = A^-1(B)

For a linear system in several variables that is dependent, how do you write the generic ordered triplet solution?

Say that after getting the equations in triangular form, you have an equation in x, y, and z, a second equation in y and z, and a third equation such as 3 = 3. Set z = t, then back substitute to get y and x values in terms of t. to get an ordered triplet.

How do you find the area of a triangle using determinants?

Make a 3×3 matrix, with column 1 being the x-coordinates, column 2 being the y-coordinates, and column 3 being 1s. The area of the triangle is +/-(1/2) × the determinant of the matrix.

How do you find the determinant of a square matrix?

Multiply each element of a row or column by its cofactor, and add the results. To make it easier, look for a row or column with lots of zeros. To make it easier, you can do row and column transformations to get zeros in a row or column.

How do you write the solution for a dependent system?

Set x = t, and then using t in one of the equations solve for y. You will then have an ordered pair: (x, the value determined for y from the equation).

What do you get when you multiply a square matrix A by its inverse A^-1, or vice versa?

The identity matrix. Note that even though multiplication is not commutative, you CAN reverse the order when multiplying a matrix by its inverse.

Gaussian Elimination

Pick 2 of the 3 equations, and manipulate to eliminate a variable from one of them. You still have 2 equations with 3 variables; eliminate the same variable from one of these 2. Next, from the 2 equations in 2 variables, eliminate a variable from one of them. The system is now in triangular form, and you can back substitute to solve. Check that the answer solves all 3 equations.

What is a minor of a matrix?

The minor of an element of a matix is the determinant of the matrix when the row and column containing that element are excluded. Mij of the element aij is the determinant of matrix obtained by deleting the ith row and the jth column of matrix A.

What are the possible solutions of a linear system in several variables?

The possibilities are the same as with a linear system in 2 variables, 1. The system may have exactly one solution. 2. The system has no solution. (inconsistent). 3. The system has infinitely many solutions (dependent).

What can determinants be used for?

They can be used to solve systems of linear equations, and can be used to determine if a matrix has an inverse.

Systems of Linear Equations: Matrices. A system of linear equations can be expressed as a rectangular array of numbers, called a matrix. A 2 by 3 matrix has how many rows and how many columns?

Two rows and three columns.

A 2×2 matrix has a b in the top row, and c d in the bottom row. What is the determinant?

ad - bc

Picture the equations for Cramer's Rule for a System of 2 Equations in 2 Variables. Cramer's rule gives an efficient way to solve systems of linear equations. But in systems with more than 3 equations, evaluating the various determinants involved is usually a long and tedious task.

js709

If a matrix is labeled "A," what is the notation for its determinant?

lAl

What is the inverse of the 2x2 matrix with a b in row 1, and c d in row 2? What is the determinant of the matrix, and when does it have no inverse?

see box on js690

Systems of 2 equations in 2 unknowns can be solved by:

substitution, elimination, or graphically. Try ex. 4, pg 639.