Evaluate each determinant in Exercises 49–52. $\begin{vmatrix}-2 & -3 & 3 & 5 \\1 & -4 & 0 & 0 \\1 & 2 & 2 & -3 \\2 & 0 & 1 & 1\end{vmatrix}$

Verified step by step guidance

1. Step 1: Write down the determinant of the 4x4 matrix as given: \[\left| \begin{array}{cccc} -2 & -3 & 3 & 5 \\ 1 & -4 & 0 & 0 \\ 1 & 2 & 2 & -3 \\ 2 & 0 & 1 & 1 \end{array} \right|\]
2. Step 2: Choose a row or column to expand the determinant. It is often easiest to pick a row or column with zeros or smaller numbers to simplify calculations. For this matrix, the second row has two zeros, so expanding along the second row is a good choice.
3. Step 3: Use the cofactor expansion formula along the chosen row. For each element \(a_{ij}\) in the row, calculate the minor determinant by removing the \(i\)th row and \(j\)th column, then multiply by \((-1)^{i+j}\) and the element \(a_{ij}\). Sum all these products to get the determinant: \[\text{det}(A) = \sum_{j=1}^4 (-1)^{2+j} a_{2j} M_{2j}\] where \(M_{2j}\) is the minor determinant after removing row 2 and column \(j\).
4. Step 4: Calculate each minor determinant \(M_{2j}\), which are 3x3 determinants. For each 3x3 matrix, you can use the rule of Sarrus or cofactor expansion again to find their values.
5. Step 5: Substitute the values of the minors back into the cofactor expansion formula, multiply by the corresponding elements and signs, then sum all terms to find the value of the original 4x4 determinant.

Key Concepts

Determinant of a Matrix

The determinant is a scalar value that can be computed from a square matrix and provides important properties such as invertibility. For a 4x4 matrix, the determinant helps determine if the matrix is singular or nonsingular, which affects solutions to systems of equations.

Expansion by Minors (Cofactor Expansion)

This method calculates the determinant of larger matrices by expanding along a row or column. It involves computing determinants of smaller submatrices (minors) multiplied by cofactors, which include a sign factor based on position.

Properties of Determinants

Determinants have properties such as linearity, the effect of row operations, and how swapping rows changes the sign. Understanding these properties can simplify determinant calculation by reducing the matrix to a simpler form.