Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)

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Question: Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)

Options:

Correct Answer: 0

Exam Year: 2022

Solution:

The determinant of G is 0 because the rows are linearly dependent.

Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)

Calculate the determinant of G = [[1, 1, 1], [1, 2, 3], [1, 3, 6]]. (2022)

Step 1: Write down the matrix G: [[1, 1, 1], [1, 2, 3], [1, 3, 6]].
Step 2: Identify the rows of the matrix: Row 1 = [1, 1, 1], Row 2 = [1, 2, 3], Row 3 = [1, 3, 6].
Step 3: Check if the rows are linearly dependent. This means we need to see if one row can be formed by a combination of the others.
Step 4: Notice that Row 3 can be formed by adding Row 1 and Row 2: [1, 1, 1] + [1, 2, 3] = [2, 3, 4], which is not Row 3. However, if we check the differences, we can see that Row 3 - Row 2 = [0, 1, 3].
Step 5: Since Row 1 is the same in all rows, we can see that the rows are dependent. This means that the determinant is 0.
Step 6: Conclude that the determinant of G is 0.

Determinant Calculation – 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 – Rows (or columns) of a matrix are linearly dependent if at least one row (or column) can be expressed as a linear combination of others, which results in a determinant of zero.