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how to calculate det A . Based on the problem solving in the picture. Answer given

To calculate the determinant of matrix \( A \) as shown in your image, let's break it down systematically based on the first matrix provided:

Matrix \( A \) is:
\[
A = \begin{bmatrix}
3 & 5 & 7 & 7 \\
1 & 4 & 7 & 2 \\
6 & 3 & 9 & 17 \\
13 & 5 & 4 & 16
\end{bmatrix}
\]

The general formula for the determinant of a \( 4 \times 4 \) matrix is computed by expansion through minors. This involves selecting one row or column and calculating the determinant of the remaining \( 3 \times 3 \) submatrices (minors).

### Step-by-Step Outline:

1. **Expand by cofactor along the first row**:
   The determinant \( \text{det}(A) \) is:
   \[
   \text{det}(A) = 3 \cdot \text{det}(A_{11}) - 5 \cdot \text{det}(A_{12}) + 7 \cdot \text{det}(A_{13}) - 7 \cdot \text{det}(A_{14})
   \]
   where \( A_{ij} \) are the \( 3 \times 3 \) submatrices formed by removing row \( i \) and column \( j \).

2. **Find each minor**:
   Each \( 3 \times 3 \) matrix, for example, \( A_{11} \) formed by removing the first row and the first column, is:
   \[
   A_{11} = \begin{bmatrix}
   4 & 7 & 2 \\
   3 & 9 & 17 \\
   5 & 4 & 16
   \end{bmatrix}
   \]
   You then compute the determinant of this \( 3 \times 3 \) matrix using the same cofactor expansion method.

3. **Repeat for each cofactor**:
   This process is repeated for \( A_{12}, A_{13}, A_{14} \), expanding them as needed and calculating the determinants of the resulting submatrices.

The handwritten notes you provided show detailed steps of a similar determinant calculation. The determinant for matrix \( A \) has already been given as 21 in the provided image.

Therefore, the determinant is:
\[
\text{det}(A) = 21
\]

Would you like a detailed breakdown of calculating the determinant for each \( 3 \times 3 \) matrix, or a walk-through of another method?

---

### Related Questions:
1. How do you calculate the inverse of matrix \( A \)?
2. What properties of determinants help in finding the inverse of a matrix?
3. How does the cofactor expansion work for larger matrices (e.g., \( 5 \times 5 \))?
4. What role does the determinant play in solving systems of linear equations?
5. Can the determinant of a matrix be used to check for singularity?

### Tip:
When computing determinants for large matrices, choosing the row or column with the most zeros simplifies the calculation!

How to calculate det A . Based on the problem solving in the picture. Answer given.

Learn how to calculate the determinant of a 4x4 matrix using cofactor expansion. This detailed step-by-step solution involves breaking down the matrix into minors and calculating the determinant for each 3x3 submatrix. Suitable for students studying Linear Algebra at the undergraduate level.

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