Chapter Five: Systems of Equations and Matrices

Question 1

independent system

Answer 1

the intersection of three planes occurs at a single point

Question 2

dependent system

Answer 2

the intersection of three planes occurs along a line (the solution is 0=0)

Question 3

inconsistent system

Answer 3

the intersection of three planes never occurs (no solution)

Question 4

matrix

Answer 4

a block of numbers, or a way of expressing a bunch of numbers or systems of equations at once

Question 5

what does the term “3 x 4” mean?

Answer 5

the matrix has 3 rows and 4 columns

Question 6

augmented matrix

Answer 6

a matrix with a bar between the ax + by and c parts of a linear equation, or a matrix that represents a system of equations in the form ax + by = c

Question 7

the number of rows in a matrix is the same as

Answer 7

the number of equations

Question 8

the number of columns in a matrix is the same as

Answer 8

the number of variables

Question 9

what are the conditions which must be met in order to add or subtract matrices?

Answer 9

1) the matrices must have the same dimension

Question 10

what are the conditions which must be met to multiply two matrices together?

Answer 10

1) the number of columns in the left-hand matrix must equal the number of rows in the right-hand matrix (the order of these cannot be flipped)

Question 11

what is the goal of the Gauss-Jordan method?

Answer 11

to obtain the following matrix:

where a = x and b = y

Question 12

what operations are allowable in the Gauss-Jordan method?

Answer 12

1) flipping the order of the 2 rows
2) multiplying through by a constant (note that multiplying by a fraction is the same as dividing by a number)
3) replacing a row with the sum of two rows

Question 13

determinant

Answer 13

a scalar that is associated with a square matrix, and is denoted by |A|

Question 14

how would you calculate the determinant of this matrix?

Answer 14

|A| = (a)(d) - (c)(b)

Question 15

what is the area of a polygon in matrix form?

Answer 15

note: list the points in counterclockwise order, and go “full circle,” including in the final matrix the last coordinate with the first coordinate

if the area = 0 the coordinates are collinear

Question 16

singular matrix

Answer 16

a matrix whose determinant is 0

Question 17

nonsingular matrix

Answer 17

a matrix whose determinant is not 0

Question 18

the identity matrix

Answer 18

has 1 along the diagonal and 0 everywhere else

a matrix times its inverse yields the identity matrix

Question 19

what types of matrices are invertible?

Answer 19

square and nonsingular matrices

Question 20

what are the steps to find the inverse of a matrix?

Answer 20

1) put in alternating signs ( + and - ) for a matrix of the same dimensions
2) block out a number and all other numbers in its row and column, and rewrite the number that is left in the new matrix (in the spot of the first number you blocked out)
3) flip the numbers along the diagonal that runs from the upper left to the lower right of the matrix
4) multiply by the inverse of the determinant ( 1 over the determinant)

Question 21

how can you find the value of x and y using matrices?

Answer 21

the matrix of xy equals the inverse of the coefficient matrix times the constant matrix

(matrix xy = (A^-1)(matrix c₁c₂)

Question 22

steps to graph inequalities:

Answer 22

1) change to y = mx + b form
2) draw the line (or other function) on the graph as a boundary (use a solid line for ≤ or ≥, and a dashed line for < or > )
3) pick a point on either side of the line and plug it into the equation
4) shade the region that works

Question 23

steps to solve linear programming problems:

Answer 23

1) write out all constraints
2) graph the constraints
3) find the vertices of the shape created by the overlapping of the constraints (note: to find the vertices use x and y-intercepts, as well as systems of equations for intersecting lines)
4) plug the values of each vertices into the profit function, or other equivalent function
note: profit cannot be negative, but other linear programming solutions can (so the minmum can be a negative number in these problem types)