Evaluate each determinant in Exercises 49–52. - 2 - 3 3 51 - 4 0 01 2 2 - 32 0 1 1

Verified step by step guidance

1. Identify the matrix as a 4x4 matrix: \( \begin{bmatrix} -2 & -3 & 3 & 5 \\ 1 & -4 & 0 & 0 \\ 1 & 2 & 2 & -3 \\ 2 & 0 & 1 & 1 \end{bmatrix} \).
2. Use the Laplace expansion (cofactor expansion) along the first row to find the determinant.
3. Calculate the determinant of the 3x3 submatrices obtained by removing the first row and each column one by one.
4. For each submatrix, use the formula for the determinant of a 3x3 matrix: \( \text{det}(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \).
5. Sum the products of the first row elements and their corresponding 3x3 determinants, considering the sign based on the position: \((-1)^{i+j} \cdot a_{ij} \cdot \text{det}(A_{ij})\).

Key Concepts

Determinant of a Matrix

The 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 (a non-zero determinant indicates invertibility) and the volume scaling factor of the linear transformation represented by the matrix. For a 2x2 matrix, the determinant is calculated as ad - bc, while for larger matrices, it can be computed using methods like cofactor expansion or row reduction.

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix by breaking it down into smaller matrices. This involves selecting a row or column and multiplying each element by its corresponding cofactor, which is the determinant of the submatrix formed by removing the row and column of that element, adjusted by a sign based on its position. This technique is particularly useful for larger matrices, allowing for a systematic approach to finding the determinant.

Row Reduction

Row reduction, or Gaussian elimination, is a process used to simplify a matrix to its row echelon form or reduced row echelon form. This technique involves performing elementary row operations, such as swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting rows. Row reduction is not only useful for solving systems of equations but also for calculating determinants, as it can simplify the matrix to a form where the determinant can be easily computed.