What is the determinant of the matrix \( E = \begin{pmatrix} 1 & 1 & 1 \

What is the determinant of the matrix \( E = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \)? (2023)

What is the determinant of the matrix \( E = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \)? (2023)

Step 1: Identify the matrix E, which is given as E = [[1, 1, 1], [1, 2, 3], [1, 3, 6]].
Step 2: Understand that the determinant of a matrix can tell us if the columns (or rows) are linearly independent.
Step 3: Check if the first column (1, 1, 1) can be formed by a combination of the other two columns (1, 2, 3) and (1, 3, 6).
Step 4: Notice that if you take the first column and subtract it from the second column, you get (0, 1, 2).
Step 5: Now, if you take the first column and subtract it from the third column, you get (0, 2, 5).
Step 6: This shows that the first column is related to the other two columns, meaning it can be expressed as a combination of them.
Step 7: Since the first column is a linear combination of the other columns, the columns are not independent.
Step 8: When the columns of a matrix are not independent, the determinant is 0.

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