If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 &a

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Question: If \\( C = \\begin{pmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\\\ 1 & 3 & 6 \\end{pmatrix} \\), find \\( \\det(C) \\).

Options:

Correct Answer: 0

Solution:

The determinant is 0 because the first column is a linear combination of the other columns.

If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), find \( \det(C) \).

If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), find \( \det(C) \).

Step 1: Identify the matrix C, which is given as C = [[1, 1, 1], [1, 2, 3], [1, 3, 6]].
Step 2: Understand that the determinant of a matrix can be zero if one column is a linear combination of the others.
Step 3: Look at the first column of the matrix, which is [1, 1, 1].
Step 4: Check if the first column can be formed by adding or scaling the other columns.
Step 5: Notice that the first column can be expressed as a combination of the second and third columns: 1 * (second column) - 1 * (third column) = [1, 1, 1].
Step 6: Since the first column is a linear combination of the other columns, the determinant of the matrix C is 0.

Determinants – The determinant of a matrix is a scalar value that can be computed from its elements and provides important properties about the matrix, such as whether it is invertible.
Linear Dependence – Columns (or rows) of a matrix are linearly dependent if at least one column can be expressed as a linear combination of the others, which results in a determinant of zero.