A = [x 3 4 b ] find x and y if A²= 3I
Answers
Step-by-step explanation:
x is -3 and y is -2.
The matrix A is
\begin{gathered}\left[\begin{array}{ccc}x&3\\y&3\\\end{array}\right]\end{gathered}
[
x
y
3
3
]
To find A², we multiply the matrix with itself.
A² = \begin{gathered}\left[\begin{array}{ccc}x&3\\y&3\\\end{array}\right]\end{gathered}
[
x
y
3
3
]
* \begin{gathered}\left[\begin{array}{ccc}x&3\\y&3\\\end{array}\right]\end{gathered}
[
x
y
3
3
]
= \begin{gathered}\left[\begin{array}{ccc}x^{2}+3y &3x+9\\xy+3y&3y+9\\\end{array}\right]\end{gathered}
[
x
2
+3y
xy+3y
3x+9
3y+9
]
The unit matrix is I = \begin{gathered}\left[\begin{array}{ccc}1&0\\0&1\end{array}\right]\end{gathered}
[
1
0
0
1
]
So, 3I would be = \begin{gathered}\left[\begin{array}{ccc}3&0\\0&3\end{array}\right]\end{gathered}
[
3
0
0
3
]
As, A² = 3I, so
\begin{gathered}\left[\begin{array}{ccc}x^{2}+3y &3x+9\\xy+3y&3y+9\\\end{array}\right]\end{gathered}
[
x
2
+3y
xy+3y
3x+9
3y+9
]
= \begin{gathered}\left[\begin{array}{ccc}3&0\\0&3\end{array}\right]\end{gathered}
[
3
0
0
3
]
3x + 9 = 0
⇒ 3x = -9
⇒ x = -3
3y + 9 = 3
⇒ 3y = -6
⇒ y = -2
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Answer: