Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 & 3

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Question: Calculate the determinant of the matrix \\( G = \\begin{pmatrix} 2 & 1 & 3 \\\\ 1 & 0 & 1 \\\\ 3 & 1 & 2 \\end{pmatrix} \\).

Options:

Correct Answer: -1

Solution:

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

Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{pmatrix} \).

Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{pmatrix} \).

Step 1: Identify the matrix G. It is given as G = [[2, 1, 3], [1, 0, 1], [3, 1, 2]].
Step 2: Write down the formula for the determinant of a 3x3 matrix. The formula is: det(G) = 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 the matrix G to the variables in the formula: a = 2, b = 1, c = 3, d = 1, e = 0, f = 1, g = 3, h = 1, i = 2.
Step 4: Calculate the first part of the formula: ei - fh = (0*2) - (1*1) = 0 - 1 = -1.
Step 5: Calculate the second part of the formula: di - fg = (1*2) - (1*3) = 2 - 3 = -1.
Step 6: Calculate the third part of the formula: dh - eg = (1*1) - (0*3) = 1 - 0 = 1.
Step 7: Substitute the calculated values back into the determinant formula: det(G) = 2*(-1) - 1*(-1) + 3*(1).
Step 8: Simplify the expression: det(G) = -2 + 1 + 3.
Step 9: Combine the numbers: det(G) = -2 + 1 = -1, and then -1 + 3 = 2.
Step 10: The final result is that the determinant of the matrix G is 2.

Determinant Calculation – The process of finding the determinant of a 3x3 matrix using the formula involving minors and cofactors.
Matrix Operations – Understanding how to perform operations on matrices, including multiplication and addition, which are often involved in determinant calculations.