To solve the given problem, follow these steps:
1. Identify the matrix:
We are given a 2x2 matrix:
[tex]\[
\begin{pmatrix}
2 & -36 \\
2 & 3
\end{pmatrix}
\][/tex]
2. Calculate the determinant:
The determinant of a 2x2 matrix
[tex]\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\][/tex]
is calculated by the formula:
[tex]\[
\text{Determinant} = ad - bc
\][/tex]
For our matrix, let's assign:
[tex]\[
a = 2, \quad b = -36, \quad c = 2, \quad d = 3
\][/tex]
Now, plug these values into the determinant formula:
[tex]\[
\text{Determinant} = (2 \times 3) - (2 \times -36)
\][/tex]
Calculate the multiplication:
[tex]\[
\text{Determinant} = 6 - (-72)
\][/tex]
Simplify the expression:
[tex]\[
\text{Determinant} = 6 + 72
\][/tex]
[tex]\[
\text{Determinant} = 78
\][/tex]
3. Find the square root of the determinant:
We have the determinant of the matrix as 78. Now, we need to find its square root:
[tex]\[
\text{Square Root} = \sqrt{78}
\][/tex]
The square root of 78 is approximately:
[tex]\[
\sqrt{78} \approx 8.831760866327848
\][/tex]
4. Present the final answer:
The determinant of the matrix is 78, and its square root is approximately 8.831760866327848.
So, the solution is:
[tex]\[
\text{Determinant: } 78, \quad \text{Square Root: } 8.831760866327848
\][/tex]
