Question7. Let and be matrices. is an matrix. The following statement is correct ( ). A. If and , then . B. If , then or . C. If , then . D. If , then .
Studdy Solution
STEP 1
What is this asking? Which statement about matrix multiplication and the identity matrix is always true? Watch out! Matrix multiplication isn't like regular multiplication, so don't assume all the same rules apply!
STEP 2
1. Analyze statement A
2. Analyze statement B
3. Analyze statement C
4. Analyze statement D
STEP 3
Statement A says: If and , then .
Is this *always* true?
Nope! Think about it: if was a regular number and *had* an inverse, we could multiply both sides by to get .
But with matrices, might not have an inverse!
STEP 4
Let's create a **counterexample**.
Let's use matrices for simplicity.
Let , , and .
Notice that .
STEP 5
Now, let's **calculate** :
STEP 6
And **calculate** :
STEP 7
So, , but !
Statement A is *false*!
STEP 8
Statement B says: If , then or .
This is also *false*!
The **identity matrix** definitely works, since .
And the **zero matrix** works, since .
But there are other matrices that work too!
STEP 9
Consider the matrix .
Then
So , but is not the zero matrix *or* the identity matrix.
Statement B is *false*!
STEP 10
Statement C says: If , then .
Let's see if we can find a **counterexample**.
STEP 11
Consider .
Then
So, , but .
Statement C is *false*!
STEP 12
Statement D says: If , then .
Let's **expand** :
STEP 13
We are given that .
So, we can **substitute** for in our expansion:
So, if , then .
Statement D is *true*!
STEP 14
The correct statement is D.
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