Matrices

Question 1

Rule 1 - Distributivity?

Answer 1

A(B+C)=AB+AC

Question 2

Rule 2 - Asociativity ?

Answer 2

(AB)C=A(BC)

Question 3

Rule 3 - Identity/zero matix multiplication ?

Answer 3

(A_mxn)(I_nxn) = A_mxn

(I_nxn)(A_mxn) = A_mxn

(Identity matrix acts as 1)

(0_pxn)(A_mxn) = 0_pxn

Question 4

Rule 4 - Transpose

Answer 4

(AB)^T = B^TA^T

Question 5

Rule 5 - Commutativity

Answer 5

AB ≠ BA

Question 6

Rule 6 - condition for matrix multiplication ?

Answer 6

no. Col of A = no. row of B

Question 7

Answer 7

Question 8

Answer 8

Question 9

Co-factor, A_ij, for the position (i,j) ?

Answer 9

A_ij = (-1)^i+j (minor of position (i,j))

or

A_ij = (-1)i+j (det of the matrix with row i, and column j removed)

Question 10

Determinant rule 1 - transpose

Answer 10

lAl = lA^Tl

Question 11

Determinant rule 2 - swapping rows or columns

Answer 11

If you swap two rows, or two columns of matrix A, to form matrix B

lAl = -lBl

Question 12

Determinant rule 3

lAl = 0 if ?

Answer 12

lAl = 0 if

two rows, or two columns, are the same or scalar multiples of each other

Question 13

Determinant rule 4 - bringing a scalar out

Answer 13

Question 14

Determinant rule 4 - extended

lλAl =

(if A is an nxn matrix)

Answer 14

lλAl = λⁿlAl

(if A is an nxn matrix)

Question 15

Determinant rule 5 - making a determinant easier to compute

Answer 15

You can add or subtract, rows or columns from each other to generate more 0’s in the determinant; this can make it easier to compute

Question 16

The inverse of a matrix condition?

Answer 16

A^-1A = I_n

AA^-1=I_n

Question 17

Inverse of a 2x2 matrix:

where lAl ≠ 0 (i.e A is “non-singular”)

Answer 17

Question 18

The adjoint of matrix ?

Answer 18

The adjoint of a matrix is the transpose of the matrix of cofactors

*REMEMBER* - the cofactors are determined by:

Aij = (-1)^i+j(det of matrix with row i and row j removed)

Question 19

The inverse of a 3x3 matrix ?

Answer 19

Question 20

Transpose of a matrix ?

Answer 20

Found by swapping rows and columns

(row 1 becomes column etc)

Question 21

Symmetric ?

Answer 21

Symmetric if:

B=B^T

Question 22

Skew-symetric ?

Answer 22

Skew-symmetric if:

B^T=-B

Question 23

Diagonal ?

Answer 23

An nxn matrix with at least one non-zero number on the main diagonal, with zeros everywhere else.

Question 24

Alien Co-factor rule?

Answer 24

Question 25

Properties that the inverse satisfy?

Answer 25

AA^-1 = I_n

Question 26

Solving systems of linear equations?

6x-3y=5

x+y=1

Answer 26

AX=B

Therefore,

X=A^-1B

Question 27

Guassian Elimination ?

(1) 2x₁ -x₂ + 3x₃ = 16
(2) -x₁ + 4x₂ - x₃ = -13
(3) x₁ +x₂ +5x₃ =19

Answer 27

Guassian Elimination: a process of solving simultaneous equations, through adding multiples of each equation together in a table to reduce variable coefficients to zero, leaving one coefficient and a solution behind.

Question 28

Finding the inverse of a 4x4 matrix ?

Answer 28

Use Gaussian Eliminiation, to transform the matrix into its identity matrix, using its identity matrix: Apply actions to both matrix and identity matrix - the matrix that the identity matrix forms is the inverse.

*(CAN ALSO USE THIS METHOD FOR ANY MATRIX)*

Question 29

Eigenvalues and Eignenvector relation?

Answer 29

For the matrix A_nxn

V is an eigenvector (nx1 column vector) with an eignenvalue λ

Providing V ≠ 0

AV=λ_V_

Question 30

Finding an eigen vector for a given matrix and eignenvalue?

Answer 30

Use (Characteristic equation).V

(A-λI_n) V = 0

*(characteristic equation in brackets, not determinant)*

Question 31

How do you find eigenvalues of a given matrix?

Answer 31

Use the characteristic equation

lA-λI_nl = 0