Chapter Five: Systems of Equations and Matrices Flashcards
independent system
the intersection of three planes occurs at a single point
dependent system
the intersection of three planes occurs along a line (the solution is 0=0)
inconsistent system
the intersection of three planes never occurs (no solution)
matrix
a block of numbers, or a way of expressing a bunch of numbers or systems of equations at once
what does the term “3 x 4” mean?
the matrix has 3 rows and 4 columns
augmented matrix
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
the number of rows in a matrix is the same as
the number of equations
the number of columns in a matrix is the same as
the number of variables
what are the conditions which must be met in order to add or subtract matrices?
1) the matrices must have the same dimension
what are the conditions which must be met to multiply two matrices together?
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)
what is the goal of the Gauss-Jordan method?
to obtain the following matrix:
where a = x and b = y
what operations are allowable in the Gauss-Jordan method?
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
determinant
a scalar that is associated with a square matrix, and is denoted by |A|
how would you calculate the determinant of this matrix?
|A| = (a)(d) - (c)(b)
what is the area of a polygon in matrix form?
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
singular matrix
a matrix whose determinant is 0
nonsingular matrix
a matrix whose determinant is not 0
the identity matrix
has 1 along the diagonal and 0 everywhere else
a matrix times its inverse yields the identity matrix
what types of matrices are invertible?
square and nonsingular matrices
what are the steps to find the inverse of a matrix?
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)
how can you find the value of x and y using matrices?
the matrix of xy equals the inverse of the coefficient matrix times the constant matrix
(matrix xy = (A-1)(matrix c1c2)
steps to graph inequalities:
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
steps to solve linear programming problems:
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)
