1 Systems of Linear Equations

Question 1

In a system of equations when are all equations satisfied?

Answer 1

At the equilibrium

Question 2

When does the substitution method become hard?

Answer 2

When there are more equations

Question 3

When is a system of equations consistent?

Answer 3

When the system has atleast one solution

Question 4

When is a system of equations inconsistent

Answer 4

When there are no solutions

Question 5

How should you rearrange equations so Gaussian elimination can be used?

Answer 5

With all the unknowns on the left and parameters on the right

Question 6

What is a matrix

Answer 6

A rectangular array of elements

Question 7

Why does an m x n matrix look like?

Answer 7

M rows and N columns

Question 8

What is a vector?

Answer 8

A matrix with only one row or one column

Question 9

What notation is used for a transposed matrix

Answer 9

A’ or A ^t if A Is the original matrix

Question 10

What does transposing a matrix do?

Answer 10

• if matrix A is m x n then matrix A’ is n x m

Question 11

What is a square matrix

Answer 11

A matrix with the same number of rows and columns

Question 12

What are symmetric matrices

Answer 12

A matrix where A=A’ this can only happen if the matrix is square

Question 13

What do you do when multiplying by a scalar?

Answer 13

Multiply all elements by the scalar

Question 14

When can matrices be added?

Answer 14

When they are the same size

Question 15

When can matrices be subtracted?

Answer 15

When they are the same size

Question 16

When can two matrices be multiplied?

Answer 16

When the number of columns of the first is equal to the number or rows of the second

Question 17

If a matrix (m x n) and a matrix (n x k) are multiplied, what are the dimensions of the result

Answer 17

m x k

Question 18

Does ABC=BAC if all letters are matrices

Answer 18

No

Question 19

Does (AB)C=A(BC) if all letters are matrices?

Answer 19

Yes

Question 20

Does A(B+C)= AB+AC if all letters are matrices?

Answer 20

Yes

Question 21

Does (AB)’=B’A’

Answer 21

Yes

Question 22

What is the identity matrix?

Answer 22

A square matrix made up of n rows and columns, where all elements are zeros except for the diagonal which are 1’s

Question 23

What happens when a matrix of the right dimensions is multiplied by the identity matrix?

Answer 23

It equals the original matrix

A x In = In x A = A

Question 24

What is the inverse matrix

Answer 24

The inverse matrix of an n x n matrix A is an n x n matrix A^-1 such that A x A^-1 = A^-1 x A = In

Question 25

When are inverse matrices defined?

Answer 25

For square matrices when the determinant doesn’t equal zero

Question 26

When is a matrix called singular

Answer 26

When the matrix doesn’t have an inverse

Question 27

When is a matrix non singular?

Answer 27

When the matrix has an inverse

Question 28

When are the inverses of matrices unique?

Answer 28

Always

Question 29

How is the inverse of a 2x2 matrix found

Answer 29

A^-1= 1/|A|x (a22 -a12)

(-a21 a11)

Question 30

What does (A^-1)^-1 equal?

Answer 30

A

Question 31

What is (A^-1)’ equal to?

Answer 31

(A’)^-1

Question 32

What is (AB)^-1 equal to? Assuming AB is invertible

Answer 32

B^-1A^-1

Question 33

What is (cA)^-1 equal to? Assuming c is a number not equal to 0

Answer 33

c^-1A^-1

Question 34

What form can a system of linear equations be put into

Answer 34

Matrices in the form Ax= b

Question 35

What is the determinant used for?

Answer 35

To find the inverse matrix or determine if it exists

Question 36

When do determinants occur?

Answer 36

Only with square numbers

Question 37

What is the determinant of a 1x1 matrix?

Answer 37

The only element in the matrix

Question 38

What is the determinant of a 2x2 matrix

Answer 38

a11a22-a12a21

Question 39

What is the sarrus rule

Answer 39

A way of finding the determinant in a 3x3 matrix

|A| = a11a22a33 + a12a23a31 + a13a21a32 - a31a22a13 - a32a23a11 - a33a21a12

Diagonal pattern

Question 40

How can the determinant be found for larger matrices?

Answer 40

They can be broken down until the determinant is easier to find with the below equation

|A|= a11|A11| - a12|A12| + … + (-1)^(1+n)a1n|A1n|

Question 41

What is the minor of an element

Answer 41

|Aij| with deleted row i and column j is called the minor of element aij

Question 42

What is the cofactor

Answer 42

The cofactor is the minor with the appropriate sign

Cij = (-1)^(i+j) |Aij|

Question 43

What is |A’| equal to?

Answer 43

|A|

Question 44

What is |AB| equal to

Answer 44

|A| x |B|

Question 45

What is C’ called?

Answer 45

The adjoint of the given matrix, it is the transposed matrix of the cofactors

Question 46

What are the two ways the inverse can be found?

Answer 46

A^-1 = 1/|A| x C’

A^-1 = 1/|A| x (-1)^(i+j) x |Aji|

Question 47

What is the way of finding the inverse of a particular element in a matrix

Answer 47

Aij^-1= 1/|A| x Cji

Question 48

What can be said about the equation system Ax=b if |A| is not equal to zero

Answer 48

There is a unique inverse A^-1

The solution x= A^-1 x b is the only solution

Question 49

What is cramers rule?

Answer 49

A way of finding the solution to an equation system

X1= |D1|/|A|

Xn = |D1|/|A|

Question 50

What is the Leontief model?

Answer 50

An input output model, it is a classical application of linear Algebra in economics

Question 51

Equation for the Leontief model

Answer 51

Xi = ai1x1 + … + ainxn + bi

• xi= total number of good i produced
• aij= units of good i needed to produce one unit of j
• aijxj = number of units of good i needed to produce xj units of good j
• bi = consumption of good i