Linear Algebra

Question 1

What is a matrix and what is it made of? What is the basic notation?

Answer 1

A rectangular array of numbers considered to be one mathematical object made up of elements.

Anm
n=rows
m=columns.

Question 2

What is Aij?

Answer 2

Aij is the element of row i and column j. Can be numbers and/or variables.

Question 3

What rules apply to addition and scalar multiplication?

Answer 3

To add matrices, they must be the same dimension.

The scalar is applied to each element to complete scalar multiplication.

Question 4

What is a vector?

Answer 4

A vector can be thought of as a nx1 matrix. Can be any number of rows but always 1 column.

Question 5

What is the inner product?

Answer 5

The inner product of a pair of vectors is the scalar. Multiply the elements of each vector then add the products to find the scalar.

Question 6

What rules apply to matrix multiplication?

Answer 6

The number of columns of matrix A must be equal to the number of rows of matrix B.

The remaining outer dimensions create the dimension of the new matrix.

Eg. A 2x3 matrix and a 3x2 matrix can be multiplied to end up with a 2x2 matrix.

Question 7

List the five special matrices

Answer 7

1) Square matrix
2) Symmetric matrix
3) Independent matrix
4) Identity matrix
5) Diagonal matrix

Question 8

What defines a square matrix?

Answer 8

Transpose A to become A’ meaning the rows of A become the columns of A’.
A=A’

Question 9

What defines a symmetric matrix?

Answer 9

It must be square. Transpose the matrix to find A=A’.

Question 10

What defines a diagonal matrix?

Answer 10

Non-zero entries occur only on the main diagonal aka whole numbers diagonally and zero everywhere else.

Question 11

What defines a identity matrix?

Answer 11

The matrix has 1’s is the diagonal as opposed to 0’s. The formula is IA=A.

Question 12

What defines an independant matrix?

Answer 12

It is multiplied by itself and nothing changes.

Question 13

What is Ax=b? What are we trying to find when we use this equation?

Answer 13

The linear equation of the matrix where A=matrix, x=vector, and b=vector.

Trying to find if a unique solution exists.

Question 14

What is A^-1b=x?

Answer 14

It is the inverse of Ax=b and is used to find is a unique solution exists. We use the inverse because we cannot divide matrices.

Question 15

What is a determinant used for? What does is A=0 mean?

Answer 15

To determine if a matrix (A) has an inverse (A^-1). If A=0 then A^-1 does not exist.

Question 16

List 2 things about the submatrix

Answer 16

1) The submatrix is Aij
2) Found by removing rows and columns of A to get Aij

Question 17

What is the cofator?

Answer 17

The hidden “+/-“ associated with the matrix.

Question 18

What are the steps to Cramer’s Rule?

Answer 18

1) Find determinant of A
2) Construct ancillary matrix
3) Replace vector B throughout the columns of matrix A and find determinant
4) Take the det of each new matrix and divide by det A to find the solutions for vector X

Question 19

What do A=0 and A ‘does not equal’ 0 indicate about the matrix?

Answer 19

If A=0 then A in singular and the inverse does not exist.

If A does not equal 0 then A is nonsingular and the inverse does exist.

Question 20

What are the steps for row echelon decompostion?

Answer 20

1) Write the augmented matrix
2) Perform elementary row operations such that all the elements below the diagonal are zero
3) Resulting matrix is R

Question 21

What options exist for elementary row operations?

Answer 21

1) Swap two rows
2) Multiply a row by a nonzero real number
3) Add or subtract rows

Question 22

What is the rank of the matrix? What does it mean if a matrix does not have full rank?

Answer 22

The number of linearly independent rows or columns. If a matrix is singular it does not have full rank.

Question 23

What is the row echelon form and the reduced row echelon form?

Answer 23

The row echelon form contains zeros beneath the diagonal. The reduced row echelon form contains all ones on the diagonal and zeros everywhere else.

Question 24

What is the first nonzero element called?

Answer 24

A pivot

Question 25

What is the leontif model?

Answer 25

The model describes an economy with interlinked industries. It describes that output from one industry can become an input to another industry.

Question 26

No answer req: Each industry faces an external/final demand in addition to supplying its own goods to other industries. The amount of each good is used to supply internal and external demands.

Answer 26

NA

Question 27

In the equation X=Ax + b what does each variable represent?

Answer 27

A = the matrix of technical coefficients or inputs and outputs required per industry
x = the supply vector
b = the final demand vector
X = the quantity that needs to be produced

Question 28

What are the steps to find X in the leontif matrix?

Answer 28

1) Complete reduced row echelon
2) Compute the identity matrix minus the technical coefficient matrix (I-A)
3) Find the leontif inverse (I-A)^-1 which gives X
4) Multiply x and b (the final demand vector)

Question 29

When is the leontif model “employed”? What is a sensitivity analysis?

Answer 29

To predict ripple effects.
A sensitivity analysis looks at what happens to changes in the supply vector and demand vector as each increases or decreases.

Question 30

What are Markov Chains?

Answer 30

Statistical model that describes a sequence of possible events and is applied when there are repeated observations made at regular time intervals.

Question 31

NA: Matrix of probability - columns must add up to 1

Answer 31

NA