Chapter 1.3 - 1.4 Flashcards
size of a matrix
rows X columns
What is a scalar?
constant
How do you multiply two matrices?
How do you TRANSPOSE (AT ) matrix?
interchange rows and columns
What is the trace of a matrix?
sum of the entries on the main diagonal of a SQUARE matrix
What property do all matrix traces have?
Commutative property: trace AB = trace BA
If you know that AB + BA is defined, then A and B are ____
square matrices
Matrix Properties:
A + B =
A + B = B + A
Matrix Properties:
A + (B + C) =
A + (B + C) = (A + B) + C
Matrix Properties
Associative law for multiplication:
A(BC) =
A (BC) = (AB) C
Matrix Properties
Left distributive law:
A (B + C) =
A (B + C) = AB + AC
Matrix Properties
Right distribution law:
(B + C) A =
(B + C) A = BA + CA
Properties DON’T WORK for Matrices
Not commutative:
AB ≠
AB ≠ BA
Properties DON’T WORK for Matrices
If AB = AC, then does B always = C?
NO, if AB = AC
B ≠ C
If AB = 0, then does A or B have to = 0?
NO: If AB = 0 then A or B don’t necessarily have to be = 0
What does it mean to say that two matrices COMMUTE?
AB = BA
What is an IDENTITY matrix?
square matrix with 1s on the diagonal and 0s everywhere else
A I = ?
A I = A
Can an identity matrix have an all 0 row?
NO
What is an INVERTIBLE matrix?
If AB = BA = I
B is called the inverse of A if ___
B = A-1
If A and B are invertible then ____
B = A-1 and A = B-1
What does it mean if a matrix is SINGULAR?
Not invertible
Are matrix inverses unique?
YES
if B and C are both inverses of matrix A, then _____
B = C
A A-1 =
A-1A =
A A-1 = I
A-1A = I
How do you determine if a 2x2 matrix is invertible?
Solution of a linear system by matrix inversion:
(AT)T =
(A + B)T =
(kA)T =
(AB)T =
(AT)T = A
(A + B)T = AT + BT
(kA)T = kAT
(AB)T = AT BT
if A and B are invertible matrices with the same size, then ___
AB is invertible and (AB)-1 = B-1 A-1
If A is invertible, then AT is also invertible then ___
(AT)-1 = (A-1)T
A0 =
A0 = I
If A is invertible, A-n =
A-n = (An)-1
Do the laws of exponents still work for matrices?
YES
Does (A+B)2 = A2 + 2AB + B2
(A+B)2 ≠ A2 + 2AB + B2
Because AB ≠ BA
Is a square matrix a row/column of 0s invertible?
NOT invertible → because no matrix B can be found such that BA=I
