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econ 4037 pre sessional maths > 4. Matrices > Flashcards

4. Matrices Flashcards

1
Q

For an element Amn, what does the m and n determine

A

M refers to the row of the element (height- think mountain)
N refers to column of the element (length- think Nile)

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2
Q

What is a vector?

A

A matrix with only one row or column

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3
Q

What is the trace?

A

It is the sum of the elements along the diagonal of a square matrix

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4
Q

Diagonal matrices

A

Square matrices with non zero entries only in the main diagonal and zero entries everywhere else

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5
Q

Identity matrix

A

A special type of diagonal matrix (In) with the number 1 in the main diagonal and zero everywhere else

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6
Q

Triangular matrix

A

Non zero entries in the positions above (below) the main diagonal and zero entries below (above) the main diagonal

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7
Q

Scalar

A

A matrix with a single entry (ie a real number)

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8
Q

When can you add and subtract matrices?

A

When they are of the same order

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9
Q

When can you multiply matrix Amxn and matrix Bpxq? And what dimensions will the new matrix have?

A

When n=p
The new matrix will have order m x q

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10
Q

What does B x In equal when B is a matrix?

A

B
Any matrix times by the identity matrix equals itself

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11
Q

What is a matrix transposition?

A

The rows and columns of the matrix are interchanged. It is essentially reflected in the line y=-x

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12
Q

Whet is the inverse matrix?

A

It is the matrix which satisfies the condition A x A^-1= A^-1A=In

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13
Q

What is 1/|A| x adj(A) equal to?

A

The inverse matrix A^-1

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14
Q

The determinant

A

|A| is a number which is a function of the entries of a square matrix

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15
Q

When can an inverse matrix be determined?

A

Only for a square matrix. Only if the determinant of the matrix doesn’t equal zero

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16
Q

What is the determinant of a matrix of order 1?

A

a11 (the only element available)

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17
Q

How is the determinant calculated?

A

By multiplying it’s elements by the corresponding cofactors and adding the products

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18
Q

Adjoint matrix

A

The transpose of the matrix obtained by replacing each element aij of matrix A with its corresponding cofactor

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19
Q

Does (AB)’ = B’A’

A

Yes

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20
Q

When is a matrix called singular?

A

When it doesn’t have an inverse

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21
Q

When are the inverses of matrices unique?

A

Always

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22
Q

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

A

B^-1 x A^-1

23
Q

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

A

c^-1 x A^-1

24
Q

What is the determinant used for?

A

To find the inverse matrix or determine if it exists

25
Q

What is the sarrus rule?

A

A way of finding the determinant in a 3x3 matrix |A|= a11a22a33 + a12a23a31 + a13a21a32- a31a22a13- a32a23a11- a33a21a12

26
Q

What is the minor of an element

A

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

27
Q

What is the cofactor of an element?

A

The cofactor is the minor with the appropriate sign Cij= (-1)^(i+j) |Aij|

28
Q

What is |A’| equal to?

A

|A|

29
Q

What is |AB| equal to?

A

|A| x |B|

30
Q

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

A

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

31
Q

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

A

There is an unique inverse A^-1. The solution x=A^-1 x b is the only solution

32
Q

When dealing with eigenvalues what is k?

A

An unknown n element column vector

33
Q

What is r

A

An unknown scalar

34
Q

When do we get the trivial solution k=0

A

When A-rI is non singular

35
Q

When do we get a non trivial solution to k?

A

When A-rI is singular so the determinant of A-rI=0

36
Q

Eigenvalues

A

The values of r that are a solution to det(A-rI)=0

37
Q

Eigenvector

A

A nonzero vector ki which is a particular solution of the equation for a particular eigenvalue ri

38
Q

Whet does the trace of matrix A equal?

A

The sum of the eigenvalues of A

39
Q

What does the product of the eigenvalues of A equal?

A

The determinant of A

40
Q

When is a quadratic form q(x) positive definite?

A

If q(x)>0 for all x=\ 0

41
Q

When is a quadratic form q(x) positive semi definite?

A

If q(x)>= for all x =\0

42
Q

When is a quadratic form q(x) negative definite?

A

If q(x)<0 for all x =\0

43
Q

When is a quadratic form q(x) negative semi definite?

A

If q(x)<=0 for all x=\0

44
Q

How can whether a quadratic form is +ve or -ve definite be checked

A

By looking at its leading principal minors I.e. the determinants of the leading principal is matrices of A

45
Q

When is a symmetric matrix positive definite?

A

If every leading principal minor is positive

46
Q

When is a symmetric matrix negative definite?

A

If the leading principal minors alternate in sign starting with negative

47
Q

When is a symmetric matrix positive semi definite?

A

Determinants of all principal submatrices >=0

48
Q

When is a symmetric matrix negative semi definite

A

Determinants of all principal submatrices of odd order <=0 and determinants of all principal submatrices of even order >=0

49
Q

Using eigenvalues when is a symmetric matrix +ve definite?

A

If all eigenvalues are +ve

50
Q

Using eigenvalues when is a symmetric matrix +ve semidefinite?

A

If all eigenvalues are non -ve

51
Q

Using eigenvalues when is a symmetric matrix -ve definite?

A

If all eigenvalues are -ve

52
Q

Using eigenvalues when is a symmetric matrix -ve semidefinite?

A

If all eigenvalues are non positive

53
Q

Using eigenvalues when is a symmetric matrix indefinite?

A

If it has a +ve and -ve eigenvalues

econ 4037 pre sessional maths (7 decks)

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