Use the determinant theorems to evaluate each determinant.

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, 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 value according to the theorems.
4. Use the properties that multiplying a row by a scalar multiplies the determinant by that scalar, and adding a multiple of one row to another does not change the determinant, to further simplify the matrix if needed.
5. After simplifying the matrix using these theorems and operations, calculate the determinant by multiplying the diagonal entries if the matrix is triangular, or by expanding along a row or column if necessary.

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 cofactor expansion or row operations, and it helps determine if a matrix is singular or nonsingular.

Determinant Theorems

Determinant theorems include rules like the effect of row swaps, scalar multiplication of rows, and row addition on the determinant value. For example, swapping two rows changes the sign of the determinant, and multiplying a row by a scalar multiplies the determinant by that scalar.

Using Row Operations to Simplify Determinants

Row operations can simplify the calculation of determinants by transforming the matrix into an easier form, such as upper triangular. Understanding which operations affect the determinant and how allows efficient evaluation without expanding all minors.