Linear Algebra Exam 1 (ch. 1)

A product of any number of invertible matrices is ___________, and the inverse of the product is the product of the inverses in the ________ _______.

***A product of any number of invertible matrices is invertible***, and the inverse of the product is the product of the inverses in the reverse order.

invertible

a matrix that has an inverse

singular

a matrix that is not invetible

matrix

a rectangular array of numbers, written inside brackets

to prove a statement as false, provide a

counterexample

main diagonal of a square matrix A

diagonal starting with element a sub 11

lower triangular

square matrix with all zero entries above the main diagonal

upper triangular

square matrix with all zero entries below the main diagonal

elements/entries of a matrix

the numbers in a matrix

properties of echelon form:

1. The RREF of a matrix is unique 2. REF's of a matrix is not unique 3. all REF's of a matrix have the same number of zero rows and the same pivot positions

a matrix is in row echelon form (ref) if

1. each row that contains a nonzero element has 1 as its first nonzero element (leading 1's) 2. all rows consisting of only zeros are together at the bottom of the matrix 3. for any two rows with leading 1's, the leading 1 of the lower row is to the right of the leading one the higher row **below each leading 1, there are only zeros**

to solve multiple systems of equations with the same coefficient matrix, you could either:

1. find the inverse of the coefficient matrix, and multiply it by all of constant column vectors individually or 2. set up a partition, and generate all solutions at once using Gauss-Jordan elimination

a matrix is in reduced row echelon form (rref) if

1. it is in row echelon form 2. any column containing a leading 1 has zeros for all other entries **below and above each leading 1 are only zeros**

3 elementary row operations

1. multiply each element in a row by a nonzero constant 2. interchange 2 rows 3. add a nonzero multiple of a row to another row

since each homogenous linear system has at least one solution, there are only 2 cases:

1. the trivial solution is its only solution 2. infinitely many solutions (including the trivial one)

the inverse of a 2x2 matrix

1/(ad - bc) * [d -b] [-c a ]

pivot column

A column that contains a pivot position

column vector/column matrix

A matrix consisting of a single column

row vector/row matrix

A matrix consisting of a single row

to prove that two expression are equal (A=B)

A=C =D =E =F =G =B therefore, A=B or start with B and work back

if A and B are invertible matrices of the same order,

AB is invertible ,and (AB)^-1= B^-1 * A^-1

to prove that A and B are inverses

AB=BA=I

B is the inverse of A means

AB=I and BA=I

the transpose of matrix A

A^T (switch rows and columns of matrix A) example: order 3x2 becomes 2x3

matrix equation

Ax=b where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants

let A be a square matrix, then If B is a square matrix such that BA=I, then _______ and If B is a square matrix such that AB=I, then _______

B=A^-1

the equivalence theorem (THE BIG THEOREM)

If A is a square matrix, 1. A is invertible 2. Ax=0 has only the trivial solution (x=0) (where x and 0 are bolded) 3. The RREF of A is the identity matrix 4. A can be expressed as a product of elementary matrices 5.Ax=b is consistent for every nx1 matrix b 6. Ax=b has exactly one solution for every nx1 matrix b

linear system (or system of linear equations)

a finite set of linear equations

zero matrix

a matrix in which every element is zero

coefficient matrix

a matrix that contains only the coefficients of a system of equations

solution of a linear system

a sequence of values for the variables s1,s2,s3,...sn that make all of the equations in the system true

general solution

a set of parametric equations that generate all solutions to a system of equations

elementary matrix (E)

a square matrix that can be obtained from the identity matrix by a SINGLE elementary row operation (one elementary row operation away from the identity)

triangular

a square matrix that is either upper or lower triangular (or both)

diagonal matrix

a square matrix with 0 entries everywhere expect possibly the main diagonal (you can have 0's on the main diagonal, but the only places where non zeros may occur is on the main diagonal)

partition of matrix

a subdivision of the matrix into smaller matrices using horizontal and/or vertical lines

a 2x2 matrix is singular if

ad-bc=0 (the denominator of 1/(ad - bc) is 0

what does infinitely many solution mean for n=3 (planes)

all three planes intersect along the same line

linear equation in the variables x1,x2,x3,...xn

an equation that can be written as a1x1+a2x2+a3x3+...+anxn=b, where the a's are constants and not all 0

how do you turn E back into the identity matrix?

by applying the inverse of the row operation

augmented matrix

coefficient matrix with the added column of solutions (add a vertical line)

If R is the RREF for a square matrix, then either R has a _____ or ________

either R has a row of zeros or it is the identity matrix

in a system of 3 linear equations in 2 variables, what does infinitely many solutions mean

every point on the line is a solution, the lines are coincident (same line) (on top of each other)

true or false matrix multiplication is commutative (AB=BA)

false

true or false we may prove a statement is true by showing an example of where it works

false

true or false the inverse of an elementary matrix is not also an elementary matrix

false --> the inverse of each elementary matrix is also an elementary matrix

true or false if two matrices have different order, addition and subtraction is defined

false -> it is undefined

true or false when using the row or column method to multiply matrices, the rows come from the second matrix, and the columns come from the first

false--> rows come from first matrix, and columns come from second example: to find the first row of AB, multiply the first row of A by the entire matrix B to find the second column of AB, multiply matrix A by the second column of matrix B

when doing gauss Jordan,

get the matrix in ref first, then work right to left above the leading 1's to get zeros

Gaussian elimination

helps write a matrix in REF

Gauss-Jordan elimination

helps write a matrix in RREF

a matrix is upper triangular if and only if aij =0 when

i > j

a homogenous linear system with more unknowns than equations has

infinitely many solutions (this does not apply to non homogenous equations)

a system of equations is consistent if

it has at least one solution

a system of equations is inconsistent if

it has no solutions

a matrix is lower triangular if and only if aij=0 when

j > i

for n=2, the graphs of linear equations are

lines

let A and B be square matrices of the same order. If AB is invertible, then A and B

must also be invertible

square matrix order

n x n

what does no solution mean for n=3 (planes)

no common intersection among the planes

homogenous linear equation

of the form a1x1+a2x2+a3x3+...+anxn=0

matrices A and B are row equivalent if

one can be obtained from another using a sequence of elementary row operations

a system of 3 linear equations in 2 variables has 3 cases:

one solution, no solution, infinitely many solutions

find p(A) for p(x)=x^2 -7 and A=[2 4] [3 2]

p(A)= A^2 - 7*I

when writing the leading variables as functions of the free variables, treat the free variables as

parameters

use __________ ___________ to state the solution set of a system with infinitely many solutions

parametric equations

for n=3, the graphs of linear equations are

planes in 3D

pivot position

position of leading 1 in ref (example: (row 1, column 2))

when you multiply a r x m matrix by a m x n matrix what is the result

r x n matrix

the inverse of a diagonal matrix is

reciprocal of the elements on the main diagonal if and only if each element on the main diagonal is non zero

to get the transpose of a square matrix,

reflect the entries about its main diagonal

If it is not possible to transform A into the identity using elementary row operations, A is

singular (not invertible)

to solve a system from its ref, turn it into a system of equations, and start at the _______ and work your way ______

start at bottom row and work up

to prove that A implies B (if A then B)

start with A. use logical steps to show B happens or start with A. Assume B is not true, then show a contradiction happens

the diagonal matrix to any power is

the elements on the main diagonal raised to that power for any positive integer power

in a system of 3 linear equations in 2 variables, what does no solution mean

the lines are parallel

submatrices

the smaller matrices created by partitions

in a system of 3 linear equations in 2 variables, what does one solution mean

the solution is the point of intersection of the 2 lines

trivial solution

the solution x1=0, x2=0 ,etc of a homogeneous equation

what does one solution mean for n=3 (planes)

there is one common intersection among the planes

two matrixes are equal if

they have the same order and all corresponding elements are equal

inversion algorithm

to find the inverse of an invertible matrix A, find the sequence of elementary row operations that change A into the identity, and perform that sequence of operations on the identity. (create an augmented with the matrix A on the left ,and the identity on the right. after RREF, the matrix on the right will be the inverse of A)

what is the trace of a square matrix A

tr(A) is the sum of the entries of its main diagonal

true or false for a system of linear equations, exactly one of the following is true: it has zero solutions; it has one solution; it has infinitely many solutions

true

true or false after transpiring a square matrix, the main diagonal remains the same

true

true or false each elementary matrix is invertible

true

true or false each homogenous linear system has the trivial solution

true

true or false elementary row operations are invertible

true

true or false matrix multiplication is undefined unless the number of columns in the first matrix is equal to the number of rows of the second matrix

true

true or false the inverse of a matrix is unique

true

true or false the trace is only defined for square matrices

true

true or false the row operation that creates an elementary matrix is what that elementary matrix does to any matrix being multiplied by it

true theorem: if the elementary matrix E is the result of performing a row operation on the identity matrix, and A is an m x n matrix, then EA is the matrix generated by that same row operation on A

the transpose of a lower triangular matrix is

upper triangular

free variables

variables corresponding to non-pivot columns

leading variables

variables corresponding to the pivot columns (columns containing leading 1s)

unknowns

variables used in a linear system

If A is an m x n matrix, and x is a n x 1 column vector, then Ax can be written as a linear combination of the column vectors of A with coefficients as the entries of

x

the matrix equation is Ax=b. how do you solve for x

x = A^-1 b

If A is an invertible non matrix, then for each nx1 matrix b, the system of equations Ax=b has exactly one solution:

x=A^-1b