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To evaluate a third-order determinant, or any 3x3 matrix's determinant, we use a specific formula. This formula systematically helps us to calculate the determinant, which is a special number that can tell us a lot about the matrix. For a 3x3 matrix structured as \( \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant \( det(A) \) is given by:\[ det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]This involves a series of multiplications and subtractions. Here's how it works:

• The term \( a(ei - fh) \) involves the first element of the first row and the determinant of the 2x2 submatrix formed by excluding the row and column of \( a \).
• Similarly, \( b(di - fg) \) and \( c(dh - eg) \) involve calculations on the other elements of the first row.
• This operation is crucial for determining properties of matrices and places the foundation for more complex matrix operations.

Matrix algebra is a branch of mathematics dealing with matrices—rectangular arrays of numbers, symbols, or expressions. Understanding matrices is essential for evaluating determinants. In matrix algebra, matrices can be added, subtracted, and multiplied by each other, and by scalars. Evaluating a matrix's determinant, like the third-order determinant here, is a fundamental operation. In 3x3 matrices, the role of determinant evaluation is significant. If you think of a matrix as representing a set of linear equations, the determinant can indicate whether a unique solution exists. If the determinant is zero, the matrix is said to be 'singular' or non-invertible, meaning it does not have an inverse and the equations may not have a unique solution. Otherwise, a non-zero determinant implies that the matrix is 'non-singular' and invertible. Through matrix algebra, determinants also enable easier computation of properties such as matrix inversion and solving systems of linear equations, making them invaluable tools in both theoretical and applied mathematics.

The evaluation of determinants involves step-by-step calculations to arrive at a final value, which can be used to infer important matrix properties. To evaluate a 3x3 determinant, start by identifying all elements of the matrix and substitute them into the determinant formula.Key steps include:

• Assign values from the given matrix to corresponding variables in the formula (...\(a = 20, b = 0, c = -15\)...).
• Perform substitutions into the determinant formula to set up the algebraic expression (...\( det(A) = 20(30 \times 40 - 1 \times -1) + ... \)...).
• Simplify expressions inside the parentheses by performing all multiplications and additions (...\(30 \times 40 = 1200\)... and add to get \( 1201 \)).
• Complete the multiplications for each term and finally add them together to achieve the determinant's value (...\( det(A) = 24020 + 2640 = 26660 \)).

Evaluating determinants is a straightforward yet powerful way to glean insights from matrices, crucial for advancements in mathematical understanding and applications across physics, engineering, and computer science.