Given matrix A = [[x, 3], [y, 3]], if A² = 3I, where I is the identity matrix of order 2, find x and y.

Given:

A = [[x,3],[y,3]]

A² = 3I

where:

I = [[1,0],[0,1]]

First calculate A²:

A² = [[x,3],[y,3]] [[x,3],[y,3]]

= [[x² + 3y, 3x + 9], [xy + 3y, 3y + 9]]

Also:

3I = [[3,0],[0,3]]

Comparing corresponding entries:

3x + 9 = 0

3x = −9

x = −3

and

3y + 9 = 3

3y = −6

y = −2

Check the first entry:

x² + 3y = (−3)² + 3(−2) = 9 − 6 = 3

Answer:

x = −3, y = −2.

The full product A² is formed and corresponding entries are equated with 3I.