Evaluate the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 &

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Question: Evaluate the determinant \\( \\begin{vmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\\\ 1 & 3 & 6 \\end{vmatrix} \\)

Options:

Correct Answer: 0

Solution:

The determinant is 0 because the first column is repeated.

Evaluate the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)

Evaluate the determinant \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)

Correct Answer: 0

Step 1: Write down the determinant you want to evaluate: \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \).
Step 2: Identify the columns of the matrix. The first column is \( \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \).
Step 3: Notice that the first column has the same value (1) in all three rows.
Step 4: Recognize that if any two columns (or rows) of a determinant are the same, the determinant is 0.
Step 5: Conclude that since the first column is repeated (it has the same value in all rows), the determinant 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.
Column Operations – The determinant can be affected by operations on its columns, such as swapping, scaling, or adding multiples of one column to another.
Linear Dependence – If any column (or row) of a matrix can be expressed as a linear combination of others, the determinant is zero, indicating that the matrix is singular.