Use the determinant theorems to evaluate each determinant. See Example 4.

Verified step by step guidance

1. Identify the size of the matrix for which you need to find the determinant. If it's a 2x2 matrix, use the formula: \( \text{det}(A) = ad - bc \) for a matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \).
2. For a 3x3 matrix, use the rule of Sarrus or the cofactor expansion method. The rule of Sarrus is applicable only for 3x3 matrices and involves summing the products of the diagonals from left to right and subtracting the products of the diagonals from right to left.
3. If using cofactor expansion for a 3x3 matrix, choose a row or column to expand along. Calculate the determinant by multiplying each element by its cofactor and summing the results.
4. For larger matrices, reduce the matrix to a simpler form using row operations or expand using cofactors until you reach a 2x2 matrix.
5. Apply the determinant properties and theorems, such as linearity, to simplify calculations. Remember that swapping two rows changes the sign of the determinant, and multiplying a row by a scalar multiplies the determinant by that scalar.

Key Concepts

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. It provides important information about the matrix, such as whether it is invertible (non-zero determinant) or singular (zero determinant). Determinants can be calculated using various methods, including expansion by minors, row reduction, or specific formulas for small matrices.

Determinant Theorems

Determinant theorems are rules and properties that simplify the calculation of determinants. Key theorems include the properties of linearity, the effect of row operations on determinants, and the relationship between determinants and matrix inverses. Understanding these theorems allows for more efficient evaluation of determinants, especially for larger matrices.

Matrix Operations

Matrix operations, such as addition, multiplication, and finding inverses, are fundamental in linear algebra. These operations are essential for manipulating matrices to compute determinants. Familiarity with how these operations affect the determinant, such as how multiplying a row by a scalar affects the determinant value, is crucial for applying determinant theorems effectively.