What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 1 \\ 0

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Question: What is the determinant of the matrix \\( \\begin{pmatrix} 1 & 2 & 1 \\\\ 0 & 1 & 0 \\\\ 2 & 3 & 1 \\end{pmatrix} \\)?

Options:

Correct Answer: 1

Solution:

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

What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \)?

What is the determinant of the matrix \( \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \)?

Step 1: Identify the matrix. The matrix is \( A = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \).
Step 2: Use the formula for the determinant of a 3x3 matrix: \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \), where the matrix is \( \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).
Step 3: Assign values from the matrix: \( a = 1, b = 2, c = 1, d = 0, e = 1, f = 0, g = 2, h = 3, i = 1 \).
Step 4: Calculate each part of the formula: \( ei - fh = 1*1 - 0*3 = 1 \), \( di - fg = 0*1 - 0*2 = 0 \), and \( dh - eg = 0*3 - 1*2 = -2 \).
Step 5: Substitute these values back into the determinant formula: \( \text{det}(A) = 1(1) - 2(0) + 1(-2) \).
Step 6: Simplify the expression: \( 1 - 0 - 2 = -1 \).
Step 7: Conclude that the determinant of the matrix is \( -1 \).

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