In Exercises 53–54, evaluate each determinant. | | 3 1| |7 0| || |- 2 3| |1 5| || || | 3 0| |9 - 6| || | 0 7| |3 5| |

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1. Identify the 2x2 matrices within the larger determinant: A = \( \begin{bmatrix} 3 & 1 \\ -2 & 3 \end{bmatrix} \), B = \( \begin{bmatrix} 7 & 0 \\ 1 & 5 \end{bmatrix} \), C = \( \begin{bmatrix} 3 & 0 \\ 0 & 7 \end{bmatrix} \), D = \( \begin{bmatrix} 9 & -6 \\ 3 & 5 \end{bmatrix} \).
2. Calculate the determinant of each 2x2 matrix: det(A) = \(3 \times 3 - (-2) \times 1\), det(B) = \(7 \times 5 - 0 \times 1\), det(C) = \(3 \times 7 - 0 \times 0\), det(D) = \(9 \times 5 - (-6) \times 3\).
3. Use the formula for the determinant of a 4x4 block matrix: det(\( \begin{bmatrix} A & B \\ C & D \end{bmatrix} \)) = det(A) \times det(D) - det(B) \times det(C).
4. Substitute the calculated determinants into the formula: det(\( \begin{bmatrix} A & B \\ C & D \end{bmatrix} \)) = det(A) \times det(D) - det(B) \times det(C).
5. Simplify the expression to find the determinant of the entire 4x4 matrix.

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, where the matrix is represented as [[a, b], [c, d]].

Properties of Determinants

Determinants have several key properties that simplify their computation and understanding. For instance, the determinant of a product of matrices equals the product of their determinants, and swapping two rows of a matrix changes the sign of the determinant. Additionally, if a matrix has a row of zeros, its determinant is zero, indicating that the matrix is singular and not invertible.

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of larger matrices by breaking them down into smaller ones. It involves selecting a row or column and expressing the determinant as a sum of products of the elements of that row or column and their corresponding cofactors, which are determinants of the smaller matrices formed by removing the selected row and column. This technique is particularly useful for 3x3 matrices and larger.