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# Math1050 CH08

### Systems of equations and inequalities

Side 1Side 2
A ___ of a system consists of all values for variables that are solutions to each equation in the system. solution
If a system has at least one solution, it is said to be ___. consistent
If a system has no solution it is said to be ___. inconsistent
A system of linear equations has lines that are either ___, ___, or ___. parallel, intersect, are coincident (the same line)
If the lines of a system of linear equations has one solution (intersection), the system is ___ and the equations are ___. consistent, independent
If the lines of a system of linear equations are ___ the system has no solution and the equations are ___. parallel, inconsistent
If the lines of a system of linear equations are coincident, the system has ___ solutions. The system is ___ and the equations are ___. infinite, consistent, dependent
In a matrix the first column represents the variable ___, the 2nd ___, and the 3rd ___. x, y, z
The matrix used to represent a system of equations is called ___. an augmented matrix
If a matrix doesn't include the numbers to the right of the = it is called ___. the coefficient matrix
___ are used to manipulate a matrix. Row operations
List the types of row operations of a matrix. 1. interchange any two rows. 2. replace a row by a NONZERO multiple of THAT row. 3. replace a row by the sum of that row and a constant NONZERO multiple of another row.
A matrix with the number one in diagonal from top left to bottom right with all entries below the '1s' being zero is called ___. row echelon form
A matrix with the number one in diagonal from top left to bottom right with ALL other entries being zero is called ___. reduced row echelon form
Using determinants to solve a system only works when ___. the number of equations equals the number of variables
Cramer's rule states that if D does not equal 0, x = ___, and y = ___. D_x/D, D_y/D
If D = 0, the system is either ___ or ___. inconsistent, has infinite solutions
When finding the determinant of a 3x3 matrix, three 2x2 determinants are formed called ___. minor determinants
For a square determinant, each determinant is multiplied by -1 raised to an exponent. This is called a ___. cofactor
The cofactor is found by ___. multiplying -1 by an exponent found by adding the number of the row and column together (row 1, column 2 would be 1+2=3. So raised to the third power.)
The cofactors of the first row of a 3x3 matrix are ___, ___, and ___. (-1)^2 or 1, (-1)^3 or -1, (-1)^4 or 1
The cofactors of the second row of a 3x3 matrix are ___, ___, and ___. (-1)^3 or -1, (-1)^4 or 1, (-1)^5 or -1
The cofactors of the third row of a 3x3 matrix are ___, ___, and ___. (-1)^4 or 1, (-1)^5 or -1, (-1)^6 or 1
T or F? You can use any row or column to find D. True. It's best to use one with a 0 to reduce work.
If D = 0 and at least one of the determinants is different from 0, then the system is ___ and the solution set is ___. inconsistent, { }
If D = 0 and all determinants are also 0, then the system is ___, the equations are ___, and the solution set is ___. consistent, dependent, all real numbers
The value of a determinant changes sign if any 2 rows (or columns) are ___. interchanged
If all entries in any row or column equal 0, the value of the determinant is ___. 0
If any 2 rows (or columns) of a determinant have corresponding entries that are equal, the value of the determinant is ___. 0
If any row (or column) of a determinant is multiplied by a nonzero number k, the value of the determinant is ___. also changed by a factor of k
If the entries of any row (or column) of a determinant are multiplied by a nonzero number k and the result is added to the corresponding entries of another row (or column), the value for the determinant ___. is unchanged
The sum or difference of 2 matrices is found by ___. adding or subtracting the corresponding entries of each to a new matrix
T or F? A 2x3 matrix and a 2x4 matrix can be added. F
T or F? The commutative property applies to addition of matrices. T
T or F? The associative property applies to addition of matrices. T
A ___ is a number that a matrix is multiplied by. scalar
A scalar multiple of a matrix is formed by ___. multiplying each entry by a scalar
k(hA) = (kh)A demonstrates the ___ property of scalar multiplication. associative
(k + h)A = kA + Ah demonstrates the ___ property of scalar multiplication. distributive
k(A + B) = kA + kB demonstrates the ___ property of scalar multiplication. distributive
A row vector R multiplied by a column vector C is defined as ___. r_1c_1 + r_2c_2 + r_3c_3...
A row vector and column vector can only be multiplied if they ___. have the same number of entries
When multiplying matrices, each entry of the product is found by ___. multiplying the corresponding row vector of the first matrix and the column vector of the second and adding together all of the products
T or F? For matrices, AB=BA. False
An identity matrix (I) is ___. A matrix with the number one in diagonal from top left to bottom right with ALL other entries being zero
Identity property, IA = ___. A
Identity property, AI = ___. A
(A^-1)A = ___. I
A(A^-1) = ___. I
T or F? Every matrix has an inverse. F
T or F? Every square matrix has an inverse. F
If a matrix has an inverse it is said to be ___ and ___ when it does not have an inverse. nonsingular, singular
What is the procedure for finding the inverse of a matrix? 1. form the matrix [A|I] 2. transform into reduced row echelon form 3. the matrix on the right is A^-1
The procedure of reducing rational expressions into the sum of simpler fractions is called ___. partial fraction decomposition
In partial fraction decomposition, the simpler fractions are called ___. partial fractions
Partial fraction decomposition only works with ___ fractions. proper
For improper fractions, ___ is used to reduce them. division
For the complex fraction P/Q, if Q has only nonrepeating linear factors, the partial fraction decomposition is of the form ___. A/(x-a) + B/(x-b)...
For the complex fraction P/Q, if Q has repeating linear factors, the partial fraction decomposition is of the form ___. A/(x-a) + B/(x-a)^2 + C/(x-a)^3...
For the complex fraction P/Q, if Q has nonrepeating irreducible quadratic factors, the partial fraction decomposition is of the form ___. ((Ax + B)/(ax^2 + bx + c))
For the complex fraction P/Q, if Q has repeating irreducible quadratic factors, the partial fraction decomposition is of the form ___. ((Ax + B)/(ax^2 + bx + c)) + ((Cx + D)/(ax^2 + bx + c)^2) + ((Ex + F)/(ax^2 + bx + c)^3)
Created by: drjolley