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Asked by: gvr
Created at: 2025-06-18 04:05:42
Final Answer:
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Let's break down the step-by-step solution as described in your instructions:
---
### **Step 1: Generate Two 3x3 Random Matrices**
Commands:
```matlab
R = 10*rand(3);
S = 10*rand(3);
```
- `rand(3)` creates a 3x3 matrix with entries between 0 and 1.
- Multiplying by 10 gives entries between 0 and 10.
---
### **Step 2: Multiplication by Columns**
Let’s denote:
- \( R \) as matrix \( A \)
- \( S \) as matrix \( B \)
#### **Extract Columns of S:**
```matlab
s1 = S(:,1); % First column
s2 = S(:,2); % Second column
s3 = S(:,3); % Third column
```
#### **Compute Each Column of C:**
- \( c_1 = R \times s_1 \)
- \( c_2 = R \times s_2 \)
- \( c_3 = R \times s_3 \)
```matlab
c1 = R * s1;
c2 = R * s2;
c3 = R * s3;
```
#### **Build Matrix C by Columns:**
```matlab
C = [c1 c2 c3];
```
---
### **Step 3: Multiplication by Rows**
#### **Extract Rows of R:**
```matlab
r1 = R(1,:); % First row
r2 = R(2,:); % Second row
r3 = R(3,:); % Third row
```
#### **Compute Each Row of Q:**
- \( q_1 = r_1 \times S \)
- \( q_2 = r_2 \times S \)
- \( q_3 = r_3 \times S \)
```matlab
q1 = r1 * S;
q2 = r2 * S;
q3 = r3 * S;
```
#### **Build Matrix Q by Rows:**
```matlab
Q = [q1; q2; q3];
```
---
### **Step 4: Compare the Results**
Both methods should yield the same result:
```matlab
disp(C)
disp(Q)
```
You can also use:
```matlab
isequal(C, Q)
```
This will return `1` (true) if the matrices are identical.
---
### **Final Answer**
The product matrix \( C = Q = R \times S \) can be computed either by:
- Multiplying by columns (building up each column by multiplying \( R \) by each column of \( S \)),
**OR**
- Multiplying by rows (building up each row by multiplying each row of \( R \) by \( S \)).
**Both methods will give the same matrix product.**
---
**Summary Table for MATLAB Steps:**
| Step | Command | Description |
|------|-------------|-------------------------------------------|
| 1 | R=10*rand(3); S=10*rand(3); | Generate random matrices |
| 2 | s1=S(:,1); s2=S(:,2); s3=S(:,3); | Extract columns of S |
| | c1=R*s1; c2=R*s2; c3=R*s3; | Multiply by columns |
| | C=[c1 c2 c3]; | Build matrix by columns |
| 3 | r1=R(1,:); r2=R(2,:); r3=R(3,:); | Extract rows of R |
| | q1=r1*S; q2=r2*S; q3=r3*S; | Multiply by rows |
| | Q=[q1; q2; q3]; | Build matrix by rows |
| 4 | isequal(C,Q) | Compare results |
---
**You have now learned two equivalent methods to compute matrix multiplication: by columns and by rows!**
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