Matrices

Question 1

what is a matrix

Answer 1

a rectangular array of numbers/ symbols/ functions… arranged into rows + columns
mxn matrix means m rows n collumns

Question 2

entry in M in row i and column j

Answer 2

Mij

Question 3

transpose

Answer 3

if M is an mxn matrix
M^t is the matrix with entires
(M^t)ij = Mji
becomes an nxm matrix
is a reflection of M in the diagonal starting top left
(A^t)^t = A

Question 4

symmetric matrix

Answer 4

M^t = M

Question 5

skew symmetric matrix

Answer 5

M^t = -M
if skew symmetric
Aii = -Aii = 0
any matrix is the sum of a symmetric and skew symmetric matrix

Question 6

zero matrix

Answer 6

all entries are 0

Question 7

identity matrix

Answer 7

In is the square nxn matrix with entries
Iij = δij =
0 i=/= j or 1 i=j

Question 8

Mmxn(R)

Answer 8

the set of all mxn matrices with real entries

Question 9

matrix addition

Answer 9

(A+B)ij = Aij + Bij

Question 10

scalar multiplication

Answer 10

(λA) ij = λ(Aij)
(-A) = (-1)A

Question 11

properties of addition

Answer 11

1) A + 0 = A = 0 + A (additive unit)
2) A + B = B + A (commutativity)
3) A + (-A) = 0 (additive inverse)
4) A + (B+C) = (A+B) + C (associativity)
prove by writing out entry by entry

Question 12

properties of scalar multiplication

Answer 12

1) 0A = 0
2) 1A = A
3) (λμ) A = λ(μA) (associativity)
4a) (λ + μ)A = λA + μA
b) λ(A + B) = λA + λB (distributivity)
prove by writing out entry by entry

Question 13

matrix multiplication

Answer 13

Mmxn(R) x Mnxp(R) -> Mmxp(R)
(AB)ij = ai1b1j + ai2b2j+…+ ainbnj
= Σaik bkj from k=1 to n

Question 14

properties of matrix multiplication

Answer 14

1) 0X = 0
2) IX = X
3) λ(XY) = (λX)Y = X(λY)
4) (XY)Z = X(YZ) (associativity)
5a) (X + X’)Y = XY + X’Y
X(Y + Y’) = XY + XY’
b) λ(X + X’) = λX + λX’
(λ + λ’)X = λX + λ’X
(distributivity)

Question 15

proof of 0X = 0

Answer 15

0ij = 0
0Xij = 0 for all ij

Question 16

proof of IX = X

Answer 16

(IX)ij = Σδik akj =aij

Question 17

proof of associativity

Answer 17

X(YZ) = (XY)Z
X mxn Y nxp Z nxq
(X(YZ))ij = Σ xik(YZkj)
= Σ xik( Σykr zrj)
= Σxik ykr zrj
for k 1…n and r 1..p
((XY)Z)ij = Σ (XY)ir zrj
= Σ (Σxik ykr) xrj
= Σ xik ykr zrj
for k 1…n and r 1..p
= (X(YZ))ij

Question 18

invertible

Answer 18

A is invertible or non-singular if there is an nxn matrix B with
AB = I = BA
B is the inverse of A = A^-1

Question 19

no inverse

Answer 19

non invertible
singular
no B exists where AB = I

Question 20

properties of inverse/ transpose

Answer 20

1) a matrix can have at most 1 inverse
2) if X and Y are invertible then XY is invertible and (XY)^-1 = Y^-1X^-1
3) (MN)^t = N^t M^t
and (λM)^t = λM^t
4) if A is invertible then A^t is invertible
(A^t)^-1 = (A^-1)^t
5) if A and B are square matrices and AB = I then BA = I

Question 21

proof that a matrix can have at most 1 inverse

Answer 21

assume 2 inverses B,B’
AB = BA = I
AB’ = B’A = I
(BA)B’ = B(AB’)
IB’ = BI
B’ = B

Question 22

prove
(A^t)^-1 = (A^-1)^t

Answer 22

(A^t)(A^-1)^t = (A^-1A)^t = I^t = I
so (A^-1)^t is inverse to A^t
so (A^-1)^t = (A^t)^-1

Question 23

transpose distributivity
(X +Y)^t

Answer 23

= X^t + Y^t

Question 24

solving equations using matrix

Answer 24

can reqrite as a matrix times (x,y,z) and use inverse to find x,y,z