For the matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), w

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Question: For the matrix \\( B = \\begin{pmatrix} 1 & 2 \\\\ 2 & 4 \\end{pmatrix} \\), what is the determinant \\( |B| \\)?

Options:

Correct Answer: 0

Solution:

The determinant is 0 because the rows are linearly dependent.

For the matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is the determinant \( |B| \)?

For the matrix \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is the determinant \( |B| \)?

Step 1: Identify the matrix B, which is given as B = [[1, 2], [2, 4]].
Step 2: Recall the formula for the determinant of a 2x2 matrix, which is |B| = ad - bc, where a, b, c, and d are the elements of the matrix arranged as follows: [[a, b], [c, d]].
Step 3: In our matrix B, a = 1, b = 2, c = 2, and d = 4.
Step 4: Substitute the values into the determinant formula: |B| = (1 * 4) - (2 * 2).
Step 5: Calculate the products: 1 * 4 = 4 and 2 * 2 = 4.
Step 6: Subtract the second product from the first: 4 - 4 = 0.
Step 7: Conclude that the determinant |B| is 0.

Determinant of a Matrix – The determinant is a scalar value that can be computed from the elements of a square matrix 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 one can be expressed as a linear combination of the others, which affects the determinant.