Det A    =(■(20&30&40@30&40&20@40&20&30))   (■(250@270@200))

Let's break down the expression and solve for \( \text{Det}(A) \).

Given the matrix \( A \):

\[
A = \begin{pmatrix}
20 & 30 & 40 \\
30 & 40 & 20 \\
40 & 20 & 30
\end{pmatrix}
\]

And a vector \( B \):

\[
B = \begin{pmatrix}
250 \\
270 \\
200
\end{pmatrix}
\]

### Step 1: Calculate the Determinant of Matrix \( A \)

The determinant of a 3x3 matrix is calculated as:

\[
\text{Det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
\]

For matrix \( A \):

\[
\text{Det}(A) = 20(40 \times 30 - 20 \times 30) - 30(30 \times 30 - 20 \times 40) + 40(30 \times 20 - 40 \times 40)
\]

Let's calculate each term:

1. \( 40 \times 30 = 1200 \)
2. \( 20 \times 30 = 600 \)
3. \( 30 \times 30 = 900 \)
4. \( 20 \times 40 = 800 \)
5. \( 30 \times 20 = 600 \)
6. \( 40 \times 40 = 1600 \)

Substitute these into the determinant formula:

\[
\text{Det}(A) = 20(1200 - 600) - 30(900 - 800) + 40(600 - 1600)
\]

\[
\text{Det}(A) = 20(600) - 30(100) + 40(-1000)
\]

\[
\text{Det}(A) = 12000 - 3000 - 40000
\]

\[
\text{Det}(A) = 12000 - 43000
\]

\[
\text{Det}(A) = -31000
\]

### Step 2: Interpret the Result

The determinant \( \text{Det}(A) = -31000 \).

This determinant is not zero, which implies that matrix \( A \) is invertible, and the system of linear equations has a unique solution.

---

Do you have any further questions or need more details?

### Relative Questions:
1. How would the system behave if the determinant were zero?
2. How does the value of the determinant affect the solutions of the system?
3. Can the determinant provide insights into the geometry of the system of equations?
4. How would you solve the system if the matrix \( A \) had a different structure?
5. What does a negative determinant indicate in terms of the transformation represented by the matrix?

### Tip:
A non-zero determinant of a matrix indicates that the corresponding linear system has a unique solution, whereas a zero determinant suggests either no solution or infinitely many solutions.

Learn how to calculate the determinant of a 3x3 matrix step-by-step. This problem involves matrix algebra and determinants, and includes a detailed example with matrix A and vector B. Suitable for advanced high school students and anyone interested in linear algebra.

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