Calculate (P+Q)(R+S). Then, calculate (P+Q)R+(P+Q)S and compare the results.
The expressions are equal.
Practice makes perfect
We want to determine whether the two expressions are equal.
(P+Q)(R+S)and(P+Q)R + (P+Q)S
Before we start calculating, let's note that the dimensions of every given matrix are the same, 2* 2. Therefore, addition and multiplication of every pair of the given matrices are possible. Also, each product will have dimensions 2* 2, which means that every operation in this exercise will be well defined. Let's evaluate each expression one at a time.
(P+Q)(R+S)
Let's start with the first expression.
(P+Q)(R+S)
First, we will calculate P+Q.
P Q
[
cc
3&4 1&2
]
+
[
cc
1&0 3& -2
]
2* 2 2* 2
Let's do it!
Finally, we can calculate (P+Q)(R+S). To multiply matrices, we use a process called the dot product. It follows a pattern of multiplying and adding terms across the rows of the first matrix and down the columns of the second one.
[
cc
a& b c&d
]
*
[
cc
e&f g&h
]
=
[
cc
a e+ b g&af+bh ce+dg&cf+dh
]
Let's apply this.
Next, let's consider the other expression.
(P+Q)R+(P+Q)S
We have already calculated P+Q in the last part!
P Q
[
cc
3&4 1&2
]
+
[
cc
1&0 3& -2
] =[
cc
4&4 4&0
]
2* 2 2* 2
Let's multiply P+Q by the matrix R.
The expressions turned out to be equal.
(P+Q)(R+S) = 4&24 4&20 = (P+Q)R+(P+Q)S
This result should not be a surprise to us. We know that matrix multiplication of square matrices has the Distributive Property. If we use this property on the expression (P+Q)(R+S), we will obtain the other expression.
