Question

69.
[ 1 2
If A = 2
1-2 v
2
*
1
is orthogonal then
​

Answer 1

Answer:

x = -2

y = 2

(Subject to a correction in the question... this makes A = 3B where B is orthogonal, not A)

Step-by-step explanation:

First, let's be clear that there is an error in this question!

The matrix A cannot be orthogonal because one requirement is that the rows must be unit vectors, but here, the first row has magnitude √(1+4+4) = 3.

However, the matrix can be a scalar multiple of an orthogonal matrix, so let's answer the "corrected" question:

Find x and y so that A=3B, where B is orthogonal (notice that this is scaling so that the first row now has magnitude 1 as required).

The second row must be orthogonal to the first row, so:

1×2 + 2×1 + 2x = 0  =>  2x = -4  =>  x = -2

The third row must also be orthogonal to the first row, so:

1×-2 + 2y + 2×-1 = 0  =>  2y = 4  =>  y = 2.

Check:

The 2nd and 3rd rows are also othogonal as 2×-2 + y - x = -4 + 2 - -2 = 0.

Every row and every column as a ±1 entry and two ±2 entries, so all columns and rows have magnitude √(1+4+4) = 3.  Consequently, the rows and columns of B have magnitude 1, as required.