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

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Question: If \\( J = \\begin{pmatrix} 1 & 2 & 1 \\\\ 0 & 1 & 0 \\\\ 2 & 1 & 3 \\end{pmatrix} \\), what is the value of the determinant?

Options:

Correct Answer: 0

Solution:

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

If \( J = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 3 \end{pmatrix} \), what is the value of the determinant?

If \( J = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 3 \end{pmatrix} \), what is the value of the determinant?

Step 1: Identify the matrix J, which is given as J = [[1, 2, 1], [0, 1, 0], [2, 1, 3]].
Step 2: Use the formula for the determinant of a 3x3 matrix: det(J) = a(ei - fh) - b(di - fg) + c(dh - eg), where the matrix is structured as follows: [[a, b, c], [d, e, f], [g, h, i]].
Step 3: Assign the values from the matrix to the variables: a = 1, b = 2, c = 1, d = 0, e = 1, f = 0, g = 2, h = 1, i = 3.
Step 4: Calculate the first part: ei - fh = (1*3) - (0*1) = 3 - 0 = 3.
Step 5: Calculate the second part: di - fg = (0*3) - (1*2) = 0 - 2 = -2.
Step 6: Calculate the third part: dh - eg = (0*1) - (1*2) = 0 - 2 = -2.
Step 7: Substitute these values back into the determinant formula: det(J) = 1(3) - 2(-2) + 1(-2).
Step 8: Simplify the expression: det(J) = 3 + 4 - 2.
Step 9: Calculate the final result: 3 + 4 = 7, and then 7 - 2 = 5.
Step 10: Conclude that the value of the determinant is 5.

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 such as multiplication and addition of matrix elements is essential.