Show:
Scalar Multiplication
Identify Elements
Commutative
Associative
Distributive
100
We want to go across rows and down columns, then switch them around. That will only work if the numbers of rows equals the numbers of columns.
Why do the matrices need to be square if we're going to consider doing both A × B and B × A?
100
[0 0] [0 0]
Which matrix, when multiplied with the matrix below, will yield the same result regardless of the order in which they're multiplied? [9 7] [-20]
100
Commutative
Which property does not apply to matrices?
100
The associative property has to do with rearranging parenthesis rather than the order in which terms are multiplied
If the commutative property doesn't apply to matrices, why does the associative property apply?
100
The distributive property with matrices is a combination of scalar multiplication and matrix addition, both of which work with matrices
Explain why the distributive property works for matrices.
200
Yes, since there are different combinations of numbers that will give the same entry in a matrix
Is it ever possible for the product of two matrices to yield the same answer, regardless of the order in which they're multiplied?
200
[3 2] [-1 9]
Which matrix, when multiplied with the matrix below, will yield the same result regardless of the order in which they're multiplied? [3 2] [-1 9]
200
Because you multiply the row of the first matrix with the column of the second; the order matters
Why is it logical that the commutative property doesn't apply to matrix multiplication?
200
[ 2 5 ] [3 25]
B= [2 0-1] c= [0 2] BC= [3 5 1] [1 4] [-2-1]
200
[-2 -3] [1 0]
[0 -1] [1 2]+[-2 0] [1 2]= BA+CA= [1 1] [0-1] [0 1] [0 -1]
300
a(A + B) + C = aA + (aB + C)
Which is an example of two matrices satisfying the associative and distributive properties? Let a be a scalar, and A, B, and C be three unique matrices.
300
[8 9] [-5 3]
Which matrix, when multiplied with the matrix below, will yield the same result regardless of the order in which they're multiplied? [8 9] [-5 3]
300
No, but only square matrices can be squared, since the number of rows needs to equal the number of columns
If matrix multiplication is not commutative, does that mean you cannot square the matrix?
300
[8 10 1] [18 20 1]
A=[1 2] B=[2 0-1] AB= [3 4] [3 5 1]
300
[0 3] [-1-2]
[1 2][0-1]+[1 2][-20] = AB+AC = [0-1][1 1] [0-1][0 1]
400
[ 4 8] [10 18]
A = [ 2 4 ] B = [ 2 2 ] A x B = [ 2 3] [ 5 6 ]
400
[3 6] [-1 9]
Which matrix, when multiplied with the matrix below, will yield the same result regardless of the order in which they're multiplied? [3 6] [-1 9]
400
You can multiply two matrices if the number of columns in the first matrix equals the number of rows in the second matrix.
When can you not multiply matrices?
400
[8 55] [18 115]
A=[1 2] B=[2 0 -1] C=[0 2] A(BC)= [3 4] [3 5 1] [1 4] [-2 -1]
400
[-2-3] [ 1-2]
FIND (B + C) A= A=[1 2] B=[0-1] C=[-2 0] [0-1] [1 1] [ 0 1]
500
[ 8 6 ] [10 12]
A = [ 4 3 ] B = [ 2 2 ] AxB= [ 2 2 ] [ 5 6 ]
500
[4 5] [-2 2]
Which matrix, when multiplied with the matrix below, will yield the same result regardless of the order in which they're multiplied? [4 5] [-2 2]
500
[0 9 -3] [-2-3 5] [0 12-4]
A= [1 0 -2] B= [0 3] AxB= [0 3 -1] [-2-1] [0 4]
500
[8 15] [18 115]
A=[1 2] B=[2 0 -1] C=[0 2] (AB)C= [3 4] [3 5 1] [1 4] [-2 -1]
500
[0 3] [-1-2]
FIND A(B + C)= A=[1 2] B=[0-1] C=[-2 0] [0-1] [1 1] [0 1]