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

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Question: If \\( F = \\begin{pmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\\\ 1 & 3 & 6 \\end{pmatrix} \\), what is the value of the determinant?

Options:

Correct Answer: 0

Solution:

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

If \( F = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), what is the value of the determinant?

If \( F = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), what is the value of the determinant?

Step 1: Identify the matrix F, which is given as F = [[1, 1, 1], [1, 2, 3], [1, 3, 6]].
Step 2: Understand that the determinant of a matrix can be calculated using various methods, including checking for linear combinations of columns.
Step 3: Look at the first column of the matrix, which is [1, 1, 1].
Step 4: Notice that the first column can be expressed as a combination of the second and third columns. Specifically, if you take the second column [1, 2, 3] and subtract the first column [1, 1, 1], you get [0, 1, 2].
Step 5: Similarly, if you take the third column [1, 3, 6] and subtract twice the first column [1, 1, 1], you get [0, 1, 4].
Step 6: Since the first column can be formed from the other two columns, it means the columns are linearly dependent.
Step 7: When a matrix has linearly dependent columns, its determinant is 0.
Step 8: Therefore, conclude that the determinant of the matrix F 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.