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 \\( |C| \\).

Options:

Correct Answer: 0

Solution:

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

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

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

Step 1: Identify the matrix C, which is given as C = [[1, 1, 1], [1, 2, 3], [1, 3, 6]].
Step 2: Understand that we need to find the determinant of the matrix C, denoted as |C|.
Step 3: Recall that a determinant can be calculated using various methods, but we will check for linear combinations first.
Step 4: Look at the first column of the matrix, which is [1, 1, 1].
Step 5: Notice that the first column can be expressed as a combination of the second and third columns. Specifically, if we take the second column [1, 2, 3] and subtract the first column [1, 1, 1], we get [0, 1, 2].
Step 6: Similarly, if we take the third column [1, 3, 6] and subtract twice the first column [1, 1, 1], we get [0, 1, 4].
Step 7: Since the first column can be formed from the other two columns, the columns of the matrix are linearly dependent.
Step 8: When the columns of a matrix are linearly dependent, the determinant of the matrix is 0.
Step 9: Therefore, we conclude that |C| = 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 – A set of vectors (or columns of a matrix) is linearly dependent if at least one vector can be expressed as a linear combination of the others, which implies that the determinant is zero.