To solve the given problem, let's break down the steps to find the determinant of the matrix [tex]\( \left|\begin{array}{cc} 3a & 3b \\ 3c & 3d \end{array}\right| \)[/tex] given that [tex]\( \left|\begin{array}{cc} a & b \\ c & d \end{array}\right| = 5 \)[/tex].
### Step-by-Step Solution
1. Understanding Determinants and Scalar Multiplication:
The determinant of a [tex]\(2 \times 2\)[/tex] matrix [tex]\( \left|\begin{array}{cc} a & b \\ c & d \end{array}\right| \)[/tex] is given by the formula:
[tex]\[
ad - bc
\][/tex]
2. Given Information:
We are given:
[tex]\[
\left|\begin{array}{cc} a & b \\ c & d \end{array}\right| = 5
\][/tex]
3. Effect of Scalar Multiplication on Determinants:
When each element of a [tex]\( n \times n \)[/tex] matrix is multiplied by a scalar [tex]\( k \)[/tex], the determinant of the new matrix is [tex]\( k^n \)[/tex] times the determinant of the original matrix. For a [tex]\(2 \times 2\)[/tex] matrix, this can be written as follows:
[tex]\[
\left|\begin{array}{cc} ka & kb \\ kc & kd \end{array}\right| = k^2 \left|\begin{array}{cc} a & b \\ c & d \end{array}\right|
\][/tex]
In our problem, each element of the matrix is multiplied by 3, so [tex]\( k = 3 \)[/tex] and [tex]\( n = 2 \)[/tex].
4. Calculating the Determinant of the New Matrix:
Using the above property:
[tex]\[
\left|\begin{array}{cc} 3a & 3b \\ 3c & 3d \end{array}\right| = 3^2 \left|\begin{array}{cc} a & b \\ c & d \end{array}\right|
\][/tex]
5. Substitute the Given Determinant:
Given [tex]\( \left|\begin{array}{cc} a & b \\ c & d \end{array}\right| = 5 \)[/tex]:
[tex]\[
\left|\begin{array}{cc} 3a & 3b \\ 3c & 3d \end{array}\right| = 3^2 \cdot 5
\][/tex]
[tex]\[
\left|\begin{array}{cc} 3a & 3b \\ 3c & 3d \end{array}\right| = 9 \cdot 5
\][/tex]
[tex]\[
\left|\begin{array}{cc} 3a & 3b \\ 3c & 3d \end{array}\right| = 45
\][/tex]
### Conclusion
The determinant of the matrix [tex]\( \left|\begin{array}{cc} 3a & 3b \\ 3c & 3d \end{array}\right| \)[/tex] is [tex]\( 45 \)[/tex].
Thus, the final answer is:
[tex]\[
45
\][/tex]
