Question

In Exercises 46–51, evaluate each determinant.

Accepted Answer

Step 1: Understand the structure of the determinant. A determinant is a scalar value that can be computed from the elements of a square matrix. For a 2x2 matrix, the determinant is calculated using the formula: det(A) = a*d - b*c, where the matrix is [[a, b], [c, d]]. For larger matrices, such as 3x3, the determinant is calculated using cofactor expansion. Step 2: Identify the size of the matrix provided in the problem. If it is a 2x2 matrix, use the formula det(A) = a*d - b*c. If it is a 3x3 matrix, use cofactor expansion along a row or column. For example, for a 3x3 matrix [[a, b, c], [d, e, f], [g, h, i]], the determinant is calculated as: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg). Step 3: Substitute the values from the matrix into the appropriate determinant formula. For a 2x2 matrix, substitute the values of a, b, c, and d. For a 3x3 matrix, substitute the values of a, b, c, d, e, f, g, h, and i into the cofactor expansion formula. Step 4: Simplify the expressions by performing the necessary multiplications and subtractions. For a 2x2 matrix, calculate the product of a*d and b*c, then subtract the second product from the first. For a 3x3 matrix, calculate each minor determinant (e.g., ei - fh, di - fg, dh - eg) and combine them with the appropriate signs. Step 5: Verify your work by double-checking the calculations for accuracy. Ensure that all substitutions and arithmetic operations are correct. If the determinant is zero, it indicates that the matrix is singular (non-invertible).

In Exercises 46–51, evaluate each determinant.

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Key Concepts

Determinants

Square Matrices

Cofactor Expansion