https://file.aiquickdraw.com/mathbot/file/1729049127604tedssgjg.jpg

The image shows a matrix, and it looks like you're being asked to find its determinant. The matrix provided is:

\[
\begin{bmatrix}
1 & 0 & 7 \\
4 & 5 & 9 \\
1 & 6 & 2
\end{bmatrix}
\]

Now, let's calculate the determinant of this 3x3 matrix using the formula:

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

Where the matrix elements are:

\[
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]

In your matrix:
- \(a = 1\), \(b = 0\), \(c = 7\)
- \(d = 4\), \(e = 5\), \(f = 9\)
- \(g = 1\), \(h = 6\), \(i = 2\)

### Step-by-step solution:

1. Compute the first term: \(a(ei - fh)\)
   \[
   1 \times (5 \times 2 - 9 \times 6) = 1 \times (10 - 54) = 1 \times (-44) = -44
   \]

2. Compute the second term: \(-b(di - fg)\)
   \[
   0 \times (4 \times 2 - 9 \times 1) = 0 \times (8 - 9) = 0
   \]
   
3. Compute the third term: \(c(dh - eg)\)
   \[
   7 \times (4 \times 6 - 5 \times 1) = 7 \times (24 - 5) = 7 \times 19 = 133
   \]

Now sum the results:
\[
\text{det}(A) = -44 + 0 + 133 = 89
\]

Thus, the determinant of the matrix is **89**.

Let me know if you'd like more details on the steps or further explanation!

### Questions:
1. How does the determinant help in understanding matrix properties?
2. What happens if the determinant of a matrix is zero?
3. Can you explain the cofactor expansion method for finding a determinant?
4. How is the determinant related to matrix inverses?
5. What are some practical applications of matrix determinants?

**Tip:** When calculating determinants for large matrices, it's useful to first check if any rows or columns can be simplified (e.g., zeros) to make the calculation easier.

Find the determinant of the matrix: 
\[ 
\begin{bmatrix}
1 & 0 & 7 \\
4 & 5 & 9 \\
1 & 6 & 2
\end{bmatrix} 
\]

Learn how to calculate the determinant of a 3x3 matrix with a detailed solution. This problem involves key linear algebra concepts such as the matrix determinant formula. Suitable for high school students (Grades 10-12).

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