Math 1229 Unit 5-8

What is the intersection when a whole bottom row is just zeros

Infinite solutions

What is the intersection when a whole bottom row is zeros and the answer side has a number

No solutions

what solution does no parameters indicate

1 point or no solutions

What solution does 1 parameter indicate

A line

What solutions does 2 parameters indicate

A plane

How do you know when there is a unique solution

Every column has a leading 1, the solution is on the RHS of the line in the matrix

What are the 3 elementary row operations

Multiply a row by a number

Swap 2 rows

Repace a row by the sum of itself and a multiple of another row

What are the criteria for RREF

1. Any zero rows (rows consisting of only zeros) is at the bottom

2. In each non zero row, the first non zero entre is 1 (leading 1)

3. In a column that contains a leading 1, all other entries are zero

4. In any two rows with some non zero entries, the leading 1 of the higher row is farther to the left (step pattern)

How do you write the subvalues of rows and columns of a matrix

A_mxn

m= rows

n=columns

Equal matrices

They have the same dimensions and their corresponding entries are the same

When can matrices be added

If they have the same dimensions

Different dimensions = undefined

Transposing a matrix

The rows and columns swap

Ex. A_mxn becomes A_nxm

Properties of the transpose

(A^T)^T = A

(A+B)^T = A^T + B^T

(cA)^T = cA^T

Matrix multiplication

The two matrices can be multiplied if the number of columns in the first matrix= number of rows in the second column

The product matrix will be the number of rows from the first matrix and number of columns from the second matrix

When can a matrix be multiplied by itself

Must be a square matrix

What does A⁰ mean

The identity matrix

Does every number have an inverse

zero is the only number that doesn’t have an inverse

Can you find the inverse of a non-square matrix

No, it must be a square matrix

If B is the inverse of A, then AB = I = BA

What is the inverse of the identity matrix

The identity matrix

How to find the inverse of a matrix

1. Form an augmented matrix so that A is on the left of the separation and the identity matrix is on the right

2. Reduce to RREF

3. If the Left side now equals the identity matrix then A is invertible and is equal to the right side

4. If the left side does not equal the identity matrix, then A is not invertible

How to determine if a matrix is invertible when there is a variable

Set up and reduce to RREF

Once in RREF, there are 2 possible cases, the unit with the variable equals zero or it doesn’t equal zero

Ex. K-2=0 into k=2, then subbing k=2 into all the k spots and solving

You determine this by subbing it into the variable spot or dividing by the variable spot to get 1

ex. Divide the k-2 row by k-2

Matrix for of a SLE

The coefficent of the matrix in one matrix times the variable(x,y,z) = the constant term

Ax=b

How to find inverse matrix of SLE

x=A^-1b