In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\(\begin{vmatrix}\)1 & 5 & 6 \\1 & 4 & 5 \\1 & 9 & 10\(\end{vmatrix}\)$

Verified step by step guidance

1. Write down the given 3x3 determinant matrix as: \[\begin{vmatrix} 1 & 5 & 6 \\ 1 & 4 & 5 \\ 1 & 9 & 10 \end{vmatrix}\]
2. Use the alternative method for evaluating third-order determinants, which involves expanding along the first row and calculating the minors. The formula is: \[\text{Determinant} = a(ei - fh) - b(di - fg) + c(dh - eg)\] where the matrix is: \[\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\]
3. Identify the elements from the matrix: - \(a = 1\), \(b = 5\), \(c = 6\) - \(d = 1\), \(e = 4\), \(f = 5\) - \(g = 1\), \(h = 9\), \(i = 10\)
4. Calculate each of the three minors: - \(M_1 = ei - fh = (4)(10) - (5)(9)\) - \(M_2 = di - fg = (1)(10) - (5)(1)\) - \(M_3 = dh - eg = (1)(9) - (4)(1)\)
5. Substitute the values back into the determinant formula: \[\text{Determinant} = a \times M_1 - b \times M_2 + c \times M_3\] This will give you the value of the determinant after performing the arithmetic.

Key Concepts

Third-Order Determinants

A third-order determinant is a scalar value calculated from a 3x3 matrix. It helps determine properties like matrix invertibility and solutions to systems of equations. The determinant is computed using specific methods such as expansion by minors or the alternative method.

Alternative Method for Evaluating Determinants

The alternative method, often called the diagonal or Sarrus' rule, is a shortcut for calculating 3x3 determinants. It involves summing the products of diagonals from left to right and subtracting the products of diagonals from right to left, simplifying the calculation process.

Matrix Representation of Systems of Equations

Matrices represent systems of linear equations compactly, where each row corresponds to an equation and each column to coefficients of variables. Understanding this helps relate determinants to the solvability and behavior of the system.