What is the determinant of the matrix \( H = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)? (2023)

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Q: What is the determinant of the matrix \( H = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)? (2023)

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

Steps: 6

Step 1: Identify the matrix H, which is given as H = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
Step 2: Understand that the determinant is a special number that can tell us about the 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 if you add the first row (1, 2, 3) and the second row (4, 5, 6), you get (5, 7, 9), which is not equal to the third row (7, 8, 9).
Step 5: However, if you look closely, you can see that the third row can be formed by adding the first two rows together in a specific way, which shows that they are dependent.
Step 6: Since the rows are linearly dependent, the determinant of the matrix is 0.