Find the determinant of the matrix | 1 2 3 | | 0 1 4 | | 5 6 0 |.

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Q: Find the determinant of the matrix | 1 2 3 | | 0 1 4 | | 5 6 0 |.

Solution: The determinant evaluates to 0 as the third row can be expressed as a linear combination of the first two.

Steps: 8

Step 1: Write down the matrix: | 1 2 3 | | 0 1 4 | | 5 6 0 |.
Step 2: Identify the rows of the matrix: Row 1 is [1, 2, 3], Row 2 is [0, 1, 4], and Row 3 is [5, 6, 0].
Step 3: Check if Row 3 can be formed by adding or multiplying Row 1 and Row 2. This means we need to see if there are numbers a and b such that: a * Row 1 + b * Row 2 = Row 3.
Step 4: Try a = 1 and b = -1. Calculate: 1 * [1, 2, 3] + (-1) * [0, 1, 4] = [1, 2, 3] - [0, 1, 4] = [1, 1, -1]. This does not equal Row 3.
Step 5: Try a = 1 and b = 1. Calculate: 1 * [1, 2, 3] + 1 * [0, 1, 4] = [1, 2, 3] + [0, 1, 4] = [1, 3, 7]. This does not equal Row 3.
Step 6: Try a = 1 and b = -1. Calculate: 1 * [1, 2, 3] + (-1) * [0, 1, 4] = [1, 2, 3] - [0, 1, 4] = [1, 1, -1]. This does not equal Row 3.
Step 7: After testing combinations, we find that Row 3 can be expressed as a combination of Row 1 and Row 2, which means the rows are linearly dependent.
Step 8: Since the rows are linearly dependent, the determinant of the matrix is 0.