Matrices

Question 1

Describe Gaussian elimination in terms of systems of equations rather than matrices

Answer 1

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

Question 2

What must a system of equations be for there to be a arbitrary value and hence infinitely many solutions?

Answer 2

The system must be homogeneous

all systems must equal zero

Question 3

How do you turn systems of equations into a matrix?

Answer 3

Take the coefficients of the variables and put them in a m by n array

Question 4

What does Mmxn(F) mean?

Answer 4

Defines all matrices of rows m and columns n over a field F

Question 5

What is meant by:

• a square matrix ?
• the identity matrix ?
• a diagonal matrix ?

Answer 5

• 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

Question 6

What are the 4 properties of matrix addition ?

Answer 6

• A+(B+C)=(A+B)+C
• A+B=B+A
• A+0(mxn)=A
• A+(-A)=0(mxn)

Question 7

What are the 4 properties of matrix scalar multiplication ?

Answer 7

• λ(A+B) = λA+λB
• (λ+μ)A = λA+μA
• λ(μA) = (λμ)A
• I A=A

Question 8

What is the definition of matrix multiplication?

Answer 8

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)

Question 9

True or False:

Matrix multiplication is communicative (AB=BA)

Answer 9

False

Question 10

True or False:

(AB)^T = A^T B^T

Answer 10

True

Question 11

What are the three types of row operations?

Answer 11

• Swap two rows
• Multiply a row by a non zero scalar
• Add to a row a scalar multiple of another row

Question 12

What is the definition of echelon form ?

Answer 12

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.

Question 13

What is the definition of reduced echelon form?

Answer 13

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

Question 14

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

Answer 14

• False

- True

Question 15

How can a matrix B be obtained from a matrix A such that A~B ?

Answer 15

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

Question 16

What is the equation for the number of variables in a matrix?

Answer 16

Number of non zero rows in REF + Number of parameters = Number of variables

Question 17

What is the definition of an invertable matrix ?

Answer 17

A is invertable if there is a B such that
AB=BA=Identity
If B exists this is denoted A^-1

Question 18

True or False :

(AB)^-1 does not equal A^-1 B^-1

Answer 18

False

Question 19

How would you find an inverse using row reduction ?

Answer 19

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

Question 20

What is the link between row operations and elementary matrices ?

Answer 20

Preforming a row operation is the same as multiplying by the corresponding elementary matrix

Question 21

If the general solution has no parameters then what can we say about the REF?

Answer 21

The REF is equal to the identity matrix

Question 22

What are the steps for finding the determinant of any n by n matrix?

Answer 22

• 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

Question 23

If a matrix A has a zero row, what can we say about the determinant ?

Answer 23

Det (A) = 0

Question 24

If a matrix A has a determinant that is non zero, what can we say about A?

Answer 24

A is invertable

Question 25

What is the definition of a symmetric matrix?

Answer 25

A matrix who’s transpose is equal to the original matrix