Question 1205106: Let A and B be the following matrices:
A rotates vectors clockwise by a right angle. B projects vectors onto a line with direction (1; 4).
I don't know how to format matrices on here, so I will write the top row (horizontal) and the bottom row. This is a 2 x 2 matrix.
Calculate the matrix AB[-4 1][1 4].
Sorry if the matrices are unclear. In latex, you are calculating $\[\mathbf{A}\mathbf{B} \begin{pmatrix} -4 & 1 \\ 1 & 4 \end{pmatrix}.\]$
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! ## Finding the Matrices A and B
**Matrix A: Clockwise Rotation by 90 Degrees**
A clockwise rotation matrix by 90 degrees is:
```
A = [[0, 1],
[-1, 0]]
```
**Matrix B: Projection onto the line with direction (7, 4)**
To find the projection matrix B, we first need to normalize the direction vector (7, 4):
```
v = (7, 4) / ||(7, 4)|| = (7/√65, 4/√65)
```
The projection matrix B onto the line with direction v is given by:
```
B = vv^T = [[49/65, 28/65],
[28/65, 16/65]]
```
## Calculating AB[-4 3][5 4]
**Step 1: Calculate AB**
```
AB = [[0, 1],
[-1, 0]] * [[49/65, 28/65],
[28/65, 16/65]]
= [[28/65, 16/65],
[-49/65, -28/65]]
```
**Step 2: Multiply AB by [-4 3][5 4]**
```
AB[-4 3][5 4] = [[28/65, 16/65],
[-49/65, -28/65]] * [[4, 15],
[20, 12]]
= [[472/65, 336/65],
[-1124/65, -804/65]]
```
Therefore, the result of the calculation is:
```
[[472/65, 336/65],
[-1124/65, -804/65]]
```
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