Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. $$\left[\begin{array}{rrrr} 3 & 6 & -5 & 4 \\ -2 & 2 & 6 & 0 \\ 1 & 1 & 2 & 0 \\ 0 & 3 & -1 & -1 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the given matrix is -20.

Step 1: Choose the column to expand along.

Choosing the last column to expand along simplifies the remainder of the calculations as this column has the most zeros. The column is \[ ^T (4,0,0,-1)\]

Step 2: Calculate the cofactors

Next, let's compute the 4x4 matrix's cofactor. The first element in the last column, 4, is multiplied by the determinant of the 3x3 matrix obtained by removing the row and column in which 4 appears:\(\left[\begin{array}{rrr} 2 & 6 & 0 \ 1 & 2 & 0 \ 3 & -1 & -1 \end{array}\right]\)The determinant of this 3x3 matrix is \(2×(2×(-1) - 0×(-1)) - 6×(1×(-1) - 0×(-1)) + 0 = -4 - (-6) = 2\)The second and the third elements of the chosen column are zeros, hence their cofactors will contribute nothing to the determinant.The last element in the last column, -1 in our case, is multiplied by the determinant of the 3x3 matrix obtained by removing the row and column in which -1 appears:\(\left[\begin{array}{rrr} 3 & 6 & -5 \ -2 & 2 & 6 \ 1 & 1 & 2 \end{array}\right]\)The determinant of this is \(3×(2×2 - 6×1) - 6×(-2×2 - 1×6) - -5×(-2×1 - 1×2) = -6 - (-24) - -10 = 28.\)

Step 3: Calculate the determinant

Lastly, sum the products calculated in the previous steps to get the determinant of given 4x4 matrix. The determinant is given by 4×(2) + 0×(cofactor of 0) + 0×(cofactor of 0) + -1×(28) = 8 - 28 = -20

Key Concepts

Cofactor Expansion

The process of cofactor expansion, also known as Laplace expansion, is a technique for calculating the determinant of a matrix. This method involves breaking down the matrix into smaller components that are easier to manage. You can choose any row or column to perform this expansion, but it is beneficial to pick the one with the most zeros. Doing so simplifies the computation as each zero will multiply the determinant of its minor by zero, therefore not contributing that part to the total sum.
In the matrix we are examining, the last column was chosen because it contains two zeros, making it an optimal choice for simplification.

4x4 Matrix

A 4x4 matrix is a square matrix that has four rows and four columns. Finding the determinant of such a matrix can seem challenging. However, using cofactor expansion makes it more manageable. The method involves creating smaller matrices by removing certain rows and columns. This breaks down the original matrix into several 3x3 matrices, from which we can easily calculate their determinants.
Therefore, by using cofactor expansion on a 4x4 matrix, you can transform it into a series of steps involving much simpler 3x3 matrices.

3x3 Matrix Determinant

The determinant of a 3x3 matrix is calculated through a more straightforward method than that of larger matrices. You can compute it using the standard formula with elements of the matrix (a, b, c, etc.). When dealing with a 3x3 matrix structured as \(\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\), the determinant is calculated as follows: \(a(ei - fh) - b(di - fg) + c(dh - eg)\).
This approach is effectively used when reducing a larger matrix into smaller parts, like what happens when performing the cofactor expansion of a 4x4 matrix.

Matrix Row and Column Removal

Matrix row and column removal is essential when calculating the determinant using cofactor expansion. When you decide to expand along a particular row or column, remove the row and column of the element you are considering. This reduced piece is called a minor, which is a smaller matrix.
Understanding this concept allows you to efficiently compute determinants of larger matrices by breaking them down into more digestible parts. This method ensures accuracy and minimizes computational errors by handling fewer operations at a time.