Describe Gaussian elimination in terms of systems of equations rather than matrices
Eliminating values of x, y, and z until you have one variable (say z) which equals a value and then work you way back up solving for the remaining variables
What must a system of equations be for there to be a arbitrary value and hence infinitely many solutions?
The system must be homogeneous
all systems must equal zero
How do you turn systems of equations into a matrix?
Take the coefficients of the variables and put them in a m by n array
What does Mmxn(F) mean?
Defines all matrices of rows m and columns n over a field F
What is meant by:
- a square matrix ?
- the identity matrix ?
- a diagonal matrix ?
- Square is where m = n
- Identity only has coefficients 1 along the main diagonal
- Diagonal is where the only entries in the matrix are along the main diagonal
What are the 4 properties of matrix addition ?
- A+(B+C)=(A+B)+C
- A+B=B+A
- A+0(mxn)=A
- A+(-A)=0(mxn)
What are the 4 properties of matrix scalar multiplication ?
- λ(A+B) = λA+λB
- (λ+μ)A = λA+μA
- λ(μA) = (λμ)A
- I A=A
What is the definition of matrix multiplication?
For A=[aij] and B=[bij] their product AB is the matrix [cij] where [cij] = Σ a[ik] b[kj]
(Take the ith row and jth column and sum them for the new entry)
True or False:
Matrix multiplication is communicative (AB=BA)
False
True or False:
(AB)^T = A^T B^T
True
What are the three types of row operations?
- Swap two rows
- Multiply a row by a non zero scalar
- Add to a row a scalar multiple of another row
What is the definition of echelon form ?
The leading non zero entry in every row is to the right of the leading entry of the previous row and any zero rows are placed at the bottom of the matrix.
What is the definition of reduced echelon form?
In addition to being in echelon form:
- All leading entries are equal to 1
- All other entries in the column of the leading entry are equal to 0
True or False:
- Every matrix can not be transformed into an echelon matrix by row operation
- Every matrix can be transformed into a reduced echelon matrix by row operations and the result will be unique
- False
- True
How can a matrix B be obtained from a matrix A such that A~B ?
B can be obtained from A such that A~B when A and B are row equivalent and if and only if they have the same reduced echelon form
What is the equation for the number of variables in a matrix?
Number of non zero rows in REF + Number of parameters = Number of variables
What is the definition of an invertable matrix ?
A is invertable if there is a B such that
AB=BA=Identity
If B exists this is denoted A^-1
True or False :
(AB)^-1 does not equal A^-1 B^-1
False
How would you find an inverse using row reduction ?
Set up a matrix with the system on the right and the identity on the right and reduce the left hand side to the identity, applying every row operation to the identity at the same time
What is the link between row operations and elementary matrices ?
Preforming a row operation is the same as multiplying by the corresponding elementary matrix
If the general solution has no parameters then what can we say about the REF?
The REF is equal to the identity matrix
What are the steps for finding the determinant of any n by n matrix?
- Pick the row you would like to expand upon
- For the first entry, multiply by the matrix of signs, take that entry and then delete the ith row and jth column and then take the determinant of the remaining matrix
- Repeat this for each entry in the row
If a matrix A has a zero row, what can we say about the determinant ?
Det (A) = 0
If a matrix A has a determinant that is non zero, what can we say about A?
A is invertable
