Unit 4 Flashcards
A blank organizes a group of numbers, or variables, using specific rules of arithmetic
matrix
A blank is a rectangular array of numbers or expressions arranged in columns and rows.
matrix
the size of a matrix is given by blank
where m is the number of rows and
n is the number of columns.
m x n
A blank is composed of a single column.
column matrix
A blank is composed of a single row.
row matrix
A blank is a matrix in which all entries are 0
zero matrix
A blank has an equal number of rows and columns. n X n
square matrix
A blank is a square matrix in which all non-diagonal entries are 0
diagonal matrix
An blank is a diagonal matrix in which all of the entries along the main diagonal are 1 and all other entries are 0
identity matrix
The transpose of ab
cd
is blank
. In general, the transpose of a square matrix is another square matrix.
ac
bd
The transpose of a matrix has as blank the rows of the original matrix.
columns
Two matrices with the same size are blank by adding the corresponding entries.
added
Two matrices with the same size are blank by subtracting the corresponding entries.
subtracted
Matrices with different sizes cannot be blank.
added
To distinguish between an ordinary real number and a matrix, the term blank refers to a real number,
scalar
A blank of a matrix is a matrix in which each entry is multiplied by a scalar.
scalar multiple
The product of a matrix A
and a scalar r
is denoted
as blank
rA
A + B = B + A
Commutative property of addition
(A + B) + C = A + (B + C)
Associative property of addition
r(sA) = (rs)A
Associative property of matrix multiplication
r(A + B) = rA + rB
Distributive property of matrix addition
(r + s)A = rA + sA
Distributive property of matrix scalar multiplication
For blank, an matrix m x n can only be multiplied by an n x p matrix to yield an m x p matrix . If the number of columns in is not equal to the number of rows in, then the product is undefined
matrix multiplication
Matrices are multiplied using the blank.
row-column rule
Matrix multiplication is not blank. AB
does not equal BA
commutative.
Similar in concept to set identities, the blank is a square matrix where all the elements of the principal diagonal are ones and all other elements are zeros. The identity matrix is denoted In
.
identify matrix
If you multiply a matrix by an identity matrix the given matrix blank. For example, m x n matrix A multiplied by In yields A.
remains unchanged
Associative property of matrix multiplication:
A(BC) = (AB)C
Left distributive property:
A(B + C)= AB + AC
Right distributive property:
(B + C) A = BA + CA
Associative property of scalars in matrix multiplication
r(AB) = (rA)B = A(rB)
Identity for matrix multiplication
ImA = AIn = A
For any three matrices changing the order of factors does not change the matrix product. You can multiply matrix B by matrix C, and then multiply the result by matrix A; or you can multiply matrix A by matrix B, and then multiply the result by matrix C.
Associative property
If matrix A is distributed from the left side, each product in the resulting sum has A on the left. (Remember, order matters in matrix multiplication.)
Left distributive property
If matrix A is distributed from the right side, each product in the resulting sum has A on the right. (Remember, order matters in matrix multiplication.)
Right distributive property
The m x m identity matrix is denoted Im. The product of any m x m matrix A and Im is always A, regardless of the order in which the multiplication was performed.
Identify for matrix multiplication
Given a square matrix A
, the matrix A^-1
is called the blank of matrix. Only square matrices can have an inverse.
inverse
Multiplying two nonsquare matrices can result in the blank.
identity matrix
Not all square matrices have an blank.
inverse
An blank (or nonsingular matrix) is a square matrix that has an inverse.
invertible matrix
A blank is a square matrix that does not have an inverse.
singular matrix
How do you find the determinant?
Multiply ad
and subtract bc
if the determinant equals 0, then blank
the inverse does not exist
if the determinant is not 0, then what is the inverse of the matrix
d -b
-c a
If matrix A is nonsingular, then blank
A^-1 is unique
A square matrix is singular if and only if what?
the determinant = 0
Any matrix that has a zero row or a zero column is blank.
singular
If A
and Bare nonsingular matrices of the same size, then the matrix AB
is blank
also nonsingular
Given a nonsingular matrix A
whose components are all integers, the inverse matrix A^-1
will contain all integer components (and no proper fractions) if and only if
blank.
the determinant = +1 or -1
A blank can be used to represent a system of linear equations, where each row represents an equation, and each column represents a different variable.
matrix
An blank is formed when two matrices are combined.
augmented matrix
The matrix A is called the blank because it consists of the coefficients of the equations.
coefficient matrix
To represent linear equations with a matrix equation, follow what steps
- To create the coefficient matrix, use the coefficients of each equation to create the rows of the coefficient matrix.
- To create the column x vector, place the variables from the equations and in the same order to create the x column.
- To create the column constant vector (column b), place the results of each equation in order in the b column.
How do you solve the system of equations by finding a vector x that satisfies the equation Ax = b
x = A^-1b
0perations performed to obtain row-equivalent matrices are called blank.
elementary row operations
Matrices are blank if a sequence of elementary row operations exists that can transform one matrix into the other. Row equivalence is sometimes denoted by the tilde (~
) symbol.
row-equivalent
Each elementary row operation performed on an blank corresponds to an operation performed on a system of linear equations.
augmented matrix
switching rows corresponds to blank.
switching equations
Multiplying a row by a blank corresponds to multiplying both sides of an equation by the same constant
non-zero constant
Adding two rows corresponds to blank.
adding two equations
An blank is a matrix that represents an elementary row operation. An elementary matrix is obtained by performing a row operation on the identity matrix.
elementary matrix
A blank is a row in a matrix that has no nonzero entries.
zero row
A blank row is a row in a matrix that has at least one nonzero entry and a nonzero column is a column in a matrix that has at least one nonzero entry.
nonzero
The blank of a row is the first nonzero entry from the left.
leading entry
A matrix is in echelon form if the matrix has what three properties:
Any zero rows are below all nonzero rows.
The leading entry of each row is to the right of the leading entries in the rows above.
All the entries below a leading entry are zeros.
A matrix is in reduced echelon form if the matrix has what properties:
Any rows that contain all zeros are below all nonzero rows.
The leading entry of each row is to the right of the leading terms in the rows above.
All the entries below a leading entry are zeros.
The leading entry for each nonzero row is 1
.
Each leading entry of 1
has zeros above. That is, the leading ones are the only nonzero entry in that column.
blank is an algorithm used to solve a system of linear equations. The algorithm consists of forward elimination and back-substitution.
Gaussian elimination
blank is the process of transforming a matrix that represents a system of linear equations into row echelon form.
Forward elimination
Five steps of Gaussian elimination: Forward elimination
Step 1: Find the current pivot of the first column. The current pivot is the first nonzero entry in a matrix column.
Step 2: If the current pivot is not in the first row, switch the first row with the row that contains the current pivot.
Step 3: Perform elementary row operations using the current pivot so that all entries below the current pivot are zeros.
Step 4: Repeat steps 1-3 for the submatrix containing all entries below and to the right of the current pivot.
Step 5: Stop when the entries below each pivot are zero.
blank is the process of substituting the value of the last variable into each of the preceding equations to obtain the values of the remaining variables.
Back-substitution
Gaussian elimination: Back-substitution - what are the steps
Step 1: Identify the corresponding system to the augmented matrix in reduced echelon form.
Step 2: Solve the last equation for the basic variable. If the system does not contain any free variables, then the basic variable has a unique value. If the system contains free variables, then write the basic variable in terms of the free variable.
Step 3: Substitute the unique value or expression for the variable into the equation above to obtain the solution for a different variable.
Step 4: Repeat steps 1-3 until a unique value or expression is found for all variables.
The blank is an algorithm that transforms a matrix into reduced echelon form. The algorithm consists of forward elimination, pivot scaling, and backward elimination.
Gauss-Jordan algorithm
Pivot scaling is the process of transforming each pivot into
what?
1
blank is the process of transforming a matrix that represents a system of linear equations from row echelon form to reduced echelon form.
Backward elimination
Gauss-Jordan elimination: Pivot scaling and backward elimination - name the steps
Step 1: Perform forward elimination.
Step 2: Scale the pivots by multiplying each row by the reciprocal of the corresponding pivot entry. This step should result in having 1s along the diagonal.
Step 3: Perform elementary row operations using the rightmost pivot so that all entries above the rightmost pivot are zeros.
Step 4: Repeat step 3 for the sub matrix containing all entries above and to the left of the rightmost pivot.
Step 5: Stop when the entries above all pivots are zero.
Elementary matrices are blank
invertible
