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

Verified step by step guidance

1. Identify the size and structure of the given determinant matrix to understand which determinant theorems can be applied effectively.
2. Recall key determinant theorems such as: the determinant of a matrix with two identical rows is zero, swapping two rows changes the sign of the determinant, multiplying a row by a scalar multiplies the determinant by that scalar, and the determinant of a triangular matrix is the product of its diagonal entries.
3. Apply row operations that simplify the matrix to a form where the determinant is easier to calculate, keeping track of how each operation affects the determinant according to the theorems.
4. Use the properties of determinants to break down the determinant calculation if the matrix can be expressed as a product or sum of simpler matrices or determinants.
5. After simplifying the matrix and applying the determinant theorems, write the determinant as a product or sum of simpler terms, then multiply or add these terms to find the determinant value.

Key Concepts

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that provides important properties such as invertibility. It can be calculated using various methods, including expansion by minors or row operations, and helps determine if a matrix is singular or nonsingular.

Determinant Theorems

Determinant theorems are rules that simplify the calculation of determinants, such as the effect of row swaps, scalar multiplication of rows, and adding multiples of one row to another. These theorems allow efficient evaluation without full expansion.

Row Operations and Their Impact on Determinants

Certain row operations change the determinant in predictable ways: swapping two rows multiplies the determinant by -1, multiplying a row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another leaves the determinant unchanged. Understanding these effects is key to using determinant theorems.