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Question 1175738: If point A = matrices 4 -4 -3 and B = 2 0 2 . Then find AB and BA
0 0 -2 2 -2 1
1 -4 -2 -1 -1 -2
I tried the following. I am not sure how to get the other numbers.
4x2 + -4x2 + -3x-1=3
0x0 + 0x-2 + -2x-1=-2
1x2 + -4x1 + -2x-2=2
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! You're on the right track with matrix multiplication! Let's break down how to calculate AB and BA step-by-step.
**Matrix Multiplication (AB)**
* **Dimensions:** First, check if the multiplication is possible. Matrix A is a 3x3 matrix, and matrix B is also a 3x3 matrix. Since the number of columns in A matches the number of rows in B, we can multiply them. The result will be a 3x3 matrix.
* **Calculation:**
* To get the element in the first row and first column of AB, you multiply the first row of A by the first column of B:
* (4 * 2) + (-4 * 2) + (-3 * -1) = 8 - 8 + 3 = 3 (You got this one right!)
* To get the element in the first row and second column of AB, you multiply the first row of A by the second column of B:
* (4 * 0) + (-4 * -2) + (-3 * -1) = 0 + 8 + 3 = 11
* To get the element in the first row and third column of AB, you multiply the first row of A by the third column of B:
* (4 * 2) + (-4 * 1) + (-3 * -2) = 8 - 4 + 6 = 10
* To get the element in the second row and first column of AB, you multiply the second row of A by the first column of B:
* (0 * 2) + (0 * 2) + (-2 * -1) = 0 + 0 + 2 = 2
* To get the element in the second row and second column of AB, you multiply the second row of A by the second column of B:
* (0 * 0) + (0 * -2) + (-2 * -1) = 0 + 0 + 2 = 2
* To get the element in the second row and third column of AB, you multiply the second row of A by the third column of B:
* (0 * 2) + (0 * 1) + (-2 * -2) = 0 + 0 + 4 = 4
* To get the element in the third row and first column of AB, you multiply the third row of A by the first column of B:
* (1 * 2) + (-4 * 2) + (-2 * -1) = 2 - 8 + 2 = -4
* To get the element in the third row and second column of AB, you multiply the third row of A by the second column of B:
* (1 * 0) + (-4 * -2) + (-2 * -1) = 0 + 8 + 2 = 10
* To get the element in the third row and third column of AB, you multiply the third row of A by the third column of B:
* (1 * 2) + (-4 * 1) + (-2 * -2) = 2 - 4 + 4 = 2
* **Result (AB):**
```
AB = | 3 11 10 |
| 2 2 4 |
| -4 10 2 |
```
**Matrix Multiplication (BA)**
* **Dimensions:** Since both B and A are 3x3 matrices, BA is also possible, and the result will be a 3x3 matrix.
* **Calculation:**
* To get the element in the first row and first column of BA, you multiply the first row of B by the first column of A:
* (2 * 4) + (0 * 0) + (2 * 1) = 8 + 0 + 2 = 10
* To get the element in the first row and second column of BA, you multiply the first row of B by the second column of A:
* (2 * -4) + (0 * 0) + (2 * -4) = -8 + 0 - 8 = -16
* To get the element in the first row and third column of BA, you multiply the first row of B by the third column of A:
* (2 * -3) + (0 * -2) + (2 * -2) = -6 + 0 - 4 = -10
* To get the element in the second row and first column of BA, you multiply the second row of B by the first column of A:
* (2 * 4) + (-2 * 0) + (1 * 1) = 8 + 0 + 1 = 9
* To get the element in the second row and second column of BA, you multiply the second row of B by the second column of A:
* (2 * -4) + (-2 * 0) + (1 * -4) = -8 + 0 - 4 = -12
* To get the element in the second row and third column of BA, you multiply the second row of B by the third column of A:
* (2 * -3) + (-2 * -2) + (1 * -2) = -6 + 4 - 2 = -4
* To get the element in the third row and first column of BA, you multiply the third row of B by the first column of A:
* (-1 * 4) + (-1 * 0) + (-2 * 1) = -4 + 0 - 2 = -6
* To get the element in the third row and second column of BA, you multiply the third row of B by the second column of A:
* (-1 * -4) + (-1 * 0) + (-2 * -4) = 4 + 0 + 8 = 12
* To get the element in the third row and third column of BA, you multiply the third row of B by the third column of A:
* (-1 * -3) + (-1 * -2) + (-2 * -2) = 3 + 2 + 4 = 9
* **Result (BA):**
```
BA = | 10 -16 -10 |
| 9 -12 -4 |
| -6 12 9 |
```
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