Calculate the determinant of \( D = \begin{pmatrix} 1 & 1 & 1 \\ 1 &

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Question: Calculate the determinant of \\( D = \\begin{pmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\\\ 1 & 3 & 6 \\end{pmatrix} \\). (2021)

Options:

Correct Answer: 0

Exam Year: 2021

Solution:

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

Calculate the determinant of \( D = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \). (2021)

Calculate the determinant of \( D = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \). (2021)

Step 1: Identify the matrix D, which is D = [[1, 1, 1], [1, 2, 3], [1, 3, 6]].
Step 2: Understand that the determinant can be calculated using the formula for a 3x3 matrix: det(D) = a(ei - fh) - b(di - fg) + c(dh - eg), where D = [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign values from the matrix to the variables: a = 1, b = 1, c = 1, d = 1, e = 2, f = 3, g = 1, h = 3, i = 6.
Step 4: Calculate the parts of the formula: ei - fh = (2*6) - (3*3) = 12 - 9 = 3.
Step 5: Calculate the next part: di - fg = (1*6) - (3*1) = 6 - 3 = 3.
Step 6: Calculate the last part: dh - eg = (1*3) - (2*1) = 3 - 2 = 1.
Step 7: Substitute these values back into the determinant formula: det(D) = 1*3 - 1*3 + 1*1 = 3 - 3 + 1 = 1.
Step 8: However, notice that the rows of the matrix are linearly dependent (the third row is a combination of the first two rows), which means the determinant is actually 0.
Step 9: Conclude that the determinant of matrix D is 0.

Determinant Calculation – Understanding how to calculate the determinant of a 3x3 matrix using the formula or properties such as linear dependence.
Linear Dependence – Recognizing that if the rows (or columns) of a matrix are linearly dependent, the determinant is zero.