What is the determinant of the matrix \( F = \begin{pmatrix} 1 & 2 & 3 \

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Question: What is the determinant of the matrix \\( F = \\begin{pmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9 \\end{pmatrix} \\)? (2020)

Options:

Correct Answer: 0

Exam Year: 2020

Solution:

The determinant is 0 because the rows are linearly dependent.

What is the determinant of the matrix \( F = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)? (2020)

What is the determinant of the matrix \( F = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)? (2020)

Step 1: Identify the matrix F, which is given as F = [[1, 2, 3], [4, 5, 6], [7, 8, 9]].
Step 2: Understand that the determinant is a special number that can be calculated from a square matrix.
Step 3: To find the determinant of a 3x3 matrix, we can use the formula: det(F) = a(ei - fh) - b(di - fg) + c(dh - eg), where the matrix is represented as: [[a, b, c], [d, e, f], [g, h, i]].
Step 4: For our matrix F, we have: a=1, b=2, c=3, d=4, e=5, f=6, g=7, h=8, i=9.
Step 5: Substitute these values into the determinant formula: det(F) = 1(5*9 - 6*8) - 2(4*9 - 6*7) + 3(4*8 - 5*7).
Step 6: Calculate each part: 5*9 = 45, 6*8 = 48, so 5*9 - 6*8 = 45 - 48 = -3.
Step 7: Calculate the second part: 4*9 = 36, 6*7 = 42, so 4*9 - 6*7 = 36 - 42 = -6.
Step 8: Calculate the third part: 4*8 = 32, 5*7 = 35, so 4*8 - 5*7 = 32 - 35 = -3.
Step 9: Now substitute these results back into the determinant formula: det(F) = 1*(-3) - 2*(-6) + 3*(-3).
Step 10: Simplify: det(F) = -3 + 12 - 9 = 0.
Step 11: Conclude that the determinant of the matrix F is 0.
Step 12: Note that a determinant of 0 indicates that the rows of the matrix are linearly dependent.

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