If \( G = \begin{pmatrix} 0 & 2 & 1 \\ 1 & 0 & 3 \\ 4 & 1 &a

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Question: If \\( G = \\begin{pmatrix} 0 & 2 & 1 \\\\ 1 & 0 & 3 \\\\ 4 & 1 & 0 \\end{pmatrix} \\), what is the determinant of G? (2020)

Options:

Correct Answer: -10

Exam Year: 2020

Solution:

Det(G) = 0 because the first column has a zero entry, leading to a linear dependence.

If \( G = \begin{pmatrix} 0 & 2 & 1 \\ 1 & 0 & 3 \\ 4 & 1 & 0 \end{pmatrix} \), what is the determinant of G? (2020)

If \( G = \begin{pmatrix} 0 & 2 & 1 \\ 1 & 0 & 3 \\ 4 & 1 & 0 \end{pmatrix} \), what is the determinant of G? (2020)

Step 1: Identify the matrix G, which is given as G = [[0, 2, 1], [1, 0, 3], [4, 1, 0]].
Step 2: Look at the first column of the matrix G. The first entry is 0.
Step 3: Understand that having a zero in the first column means that the rows of the matrix may be linearly dependent.
Step 4: Recall that if the rows of a matrix are linearly dependent, the determinant of that matrix is 0.
Step 5: Conclude that since the first column has a zero entry, the determinant of G is Det(G) = 0.

Determinant Calculation – Understanding how to compute the determinant of a 3x3 matrix using the formula or properties of determinants.
Linear Dependence – Recognizing that a column of zeros or linear dependence among rows/columns can lead to a determinant of zero.