If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |2

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Question: If \\( A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix} \\), what is \\( |2A| \\)?

Options:

Correct Answer: 8

Solution:

The determinant of \\( 2A \\) is \\( 2^2 * |A| = 4 * (-2) = -8 \\).

If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |2A| \)?

If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |2A| \)?

Step 1: Identify the matrix A, which is A = [[1, 2], [3, 4]].
Step 2: Calculate the determinant of A, denoted as |A|. The formula for the determinant of a 2x2 matrix [[a, b], [c, d]] is ad - bc.
Step 3: For matrix A, a = 1, b = 2, c = 3, d = 4. So, |A| = (1 * 4) - (2 * 3) = 4 - 6 = -2.
Step 4: Now, we need to find the determinant of 2A. The matrix 2A is obtained by multiplying each element of A by 2, resulting in 2A = [[2*1, 2*2], [2*3, 2*4]] = [[2, 4], [6, 8]].
Step 5: The determinant of 2A can be calculated using the property of determinants: |kA| = k^n * |A|, where k is a scalar and n is the size of the matrix. Here, k = 2 and n = 2 (since A is 2x2).
Step 6: Therefore, |2A| = 2^2 * |A| = 4 * (-2) = -8.

Determinant of a Matrix – Understanding how to calculate the determinant of a matrix and the effect of scalar multiplication on it.
Properties of Determinants – Applying the property that states the determinant of a scalar multiplied by a matrix is equal to the scalar raised to the power of the matrix size times the determinant of the matrix.