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Math1050 CH08
Systems of equations and inequalities
| Side 1 | Side 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
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