If \( B = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 &a

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Question: If \\( B = \\begin{pmatrix} 1 & 2 & 1 \\\\ 0 & 1 & 0 \\\\ 2 & 3 & 1 \\end{pmatrix} \\), what is \\( |B| \\)?

Options:

Correct Answer: 2

Solution:

The determinant is calculated as 1(1*1 - 0*3) - 2(0*1 - 1*2) + 1(0*3 - 1*2) = 1 - 4 - 2 = -5.

If \( B = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \), what is \( |B| \)?

If \( B = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \), what is \( |B| \)?

Correct Answer: -5

Step 1: Identify the matrix B. It is given as B = [[1, 2, 1], [0, 1, 0], [2, 3, 1]].
Step 2: Write down the formula for the determinant of a 3x3 matrix. The formula is: |B| = a(ei - fh) - b(di - fg) + c(dh - eg), where the matrix is: [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign the values from matrix B to the variables in the formula: a = 1, b = 2, c = 1, d = 0, e = 1, f = 0, g = 2, h = 3, i = 1.
Step 4: Calculate the first part of the formula: ei - fh = 1*1 - 0*3 = 1.
Step 5: Calculate the second part of the formula: di - fg = 0*1 - 1*2 = -2.
Step 6: Calculate the third part of the formula: dh - eg = 0*3 - 1*2 = -2.
Step 7: Substitute these values back into the determinant formula: |B| = 1(1) - 2(-2) + 1(-2).
Step 8: Simplify the expression: |B| = 1 + 4 - 2.
Step 9: Calculate the final result: |B| = 1 + 4 - 2 = 3.

Determinant Calculation – The question tests the ability to compute the determinant of a 3x3 matrix using the standard formula.
Matrix Operations – Understanding how to perform operations on matrix elements to find the determinant.