Calculate (P+Q)I. Then, calculate PI+QI and compare the results.
The expressions are equal.
Practice makes perfect
We want to determine whether the two expressions are equal.
(P+Q)IandPI+QI
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)I
Let's start with the first expression.
P Q I
([
cc
3&4 1&2
]
+
[
cc
1&0 3& -2
])
*
[
cc
1&0 0&1
]
2* 2 2* 2 2* 2
We need to calculate P+Q first.
Now let's multiply the resulting matrix above by I. 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 calculate it!
Next, let's consider the other expression.
P I
[
cc
3&4 1&2
]
*
[
cc
1&0 0&1
]
2* 2 2* 2
+
Q I
[
cc
1&0 3& -2
]
*
[
cc
1&0 0&1
]
2* 2 2* 2
Let's calculate PI first.
The expressions turned out to be equal.
(P+Q)I= 4&4 4&0 =PI+QI
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)I, we will obtain the other expression.
Therefore, as long as the matrices P, Q, and I are square matrices of the same dimensions, the two given expressions will be always equal.
Alternative Solution
Identity matrix
I is an identity matrix. When the identity matrix I is multiplied by another matrix A, the result is the same matrix A.
I* A=A
Let's use that fact to simplify the first expression.
(P+Q)I ⇔ P+Q
We can simplify the other expression in a similar way.
PI+QI ⇔ P+Q
Both expressions are equal to P+Q.
