Matrix Operations Flashcards
Commutative property of matrix addition
A+B=B+A
Associative property of matrix addition
(A+B)+C=A+(B+C)
Additive identity of matricies
A+0(m,n)=A, 0(m,n) is an mxn matrix where all elements are zero
Distributive properties of scalar multiplication
c(A+B)=cA+cB, (c+d)A=cA+dA
Associative property of scalar multiplication in matrices
c(dA)=(cd)A
If cA=the zero matrix, then?
Either c is zero or A is the 0 matrix
m×n n×p makes?
An m×p matrix, the inner dimensions have to be the same
Does AB=AC imply that B and C are the same?
No no no! Absolutely not!
If AB=the zero matrix, does that mean that atleast A or B are the zero matrix?
Nope! You can get the zero matrix from two non zero matricies
Associative property of matrix multiplication
(AB)C=A(BC)
Left distributive property of matrix multiplication
A(B+C)=AB+AC
Right distributive property of matrix multiplication
(A+B)C=AC+BC
c(AB) is equivalent to…
(cA)B and A(cB)
(AT)T=
A
(A+B)T=
AT+BT
(cA)T=
c(AT)
(AB)T=
BTAT (sock and shoe
What is a symmetric matrix?
When A=AT
The inverse of [a,b]
[c,d] is?
1/(ad-bc) * [d,-b]
[-c,a]
How can u figure out if a 3x3 or higher matrix is invertible?
You should be able to get the identity matrix on the left hand side if u rref the augmented matrix [A | I]
Whats a simple way to figure out if a 2x2 matrix is invertible?
It’s invertible only if the determinant is not 0.
If A is invertible, then is A^-1 invertible?
Yes. (A^-1)^-1=A
If A and B are both square matricies of the same dimension and are both invertible, then is AB invertible?
Yes. (AB^-1)=B^-1A^-1
If A is invertible, is it’s transpose also invertible?
Yes. (AT)^-1=(A^-1)T
