What is the value of the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)?

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Q: What is the value of the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)?

Solution: The determinant of the matrix is 0 because the rows are linearly dependent.

Steps: 6

Step 1: Identify the matrix. The matrix is \( A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \).
Step 2: Understand what a determinant is. The determinant is a special number that can be calculated from a square matrix.
Step 3: Check if the rows of the matrix are linearly dependent. This means that one row can be made by adding or multiplying the other rows.
Step 4: Notice that the third row (7, 8, 9) can be formed by adding the first row (1, 2, 3) and the second row (4, 5, 6).
Step 5: Since the rows are linearly dependent, the determinant of the matrix is 0.
Step 6: Conclude that the value of the determinant of the matrix is 0.