Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each determinant. $$\operatorname{det}\left[\begin{array}{rrrr}4 & 5 & -1 & -1 \\\2 & -3 & 1 & 0 \\\\-5 & 1 & 3 & 9 \\\0 & -2 & 1 & 5\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -281.

Step 1: Identify the Matrix

We are given a 4x4 matrix,\[\begin{bmatrix} 4 & 5 & -1 & -1 \ 2 & -3 & 1 & 0 \ -5 & 1 & 3 & 9 \ 0 & -2 & 1 & 5 \end{bmatrix}\]

Step 2: Choose a Row or Column for Expansion

Typically, choosing a row or column with the most zeros simplifies calculations. We'll expand along the fourth row: \([0, -2, 1, 5]\).

Step 3: Calculate Cofactors

To find the determinant, calculate cofactors for each element of the chosen row:- Cofactor of \(0\) is 0, so it contributes nothing.- Cofactor of \(-2\) involves\[\operatorname{det}\left(\begin{array}{ccc}4 & 5 & -1 \ 2 & -3 & 1 \ -5 & 1 & 3\end{array}\right)\]- Cofactor of \(1\) involves\[\operatorname{det}\left(\begin{array}{ccc}4 & 5 & -1 \ 2 & -3 & 0 \ -5 & 1 & 9\end{array}\right)\]- Cofactor of \(5\) involves\[\operatorname{det}\left(\begin{array}{ccc}4 & 5 & -1 \ 2 & -3 & 1 \ -5 & 1 & 3\end{array}\right)\]

Step 4: Compute Each 3x3 Determinant

Each 3x3 determinant is calculated using the smaller matrices derived in Step 3:1. For \(-2\) multiplier:\[\operatorname{det}\left(\begin{array}{ccc}4 & 5 & -1 \ 2 & -3 & 1 \ -5 & 1 & 3\end{array}\right) = 4(-3 \cdot 3 - 1 \cdot 1) - 5(2 \cdot 3 - 1 \cdot -5) -1(2 \cdot 1 + 5 \cdot -3)= 4(-10) - 5(-1) - (-13)= -40 + 5 + 13 = -22\]2. For \(1\) multiplier:\[\operatorname{det}\left(\begin{array}{ccc}4 & 5 & -1 \ 2 & -3 & 0 \ -5 & 1 & 9\end{array}\right) = 4(-3 \cdot 9 - 0 \cdot 1) - 5(2 \cdot 9 - 0 \cdot -5) -1(2 \cdot 1 + (-5) \cdot 3)= 4(-27) - 5(18) - (17)= -108 - 90 - 17 = -215\]3. For \(5\) multiplier (same minor as \(-2\)):\[= -22\]

Step 5: Combine Cofactor Products

Apply the cofactor products to get the determinant, \[\text{{det}} = (-1)^{4+2} \cdot -2 \cdot -22 + (-1)^{4+3} \cdot 1 \cdot -215 + (-1)^{4+4} \cdot 5 \cdot -22\]Simplify:- For \(-2\): \(2 \cdot 22 = 44\)- For \(1\): \(-215\)- For \(5\): \(-110\)Combine them,44 - 215 - 110 = -281.

Key Concepts

Understanding the 4x4 Matrix

A 4x4 matrix is a square matrix with four rows and four columns. Each element in a matrix is typically a number, and the determinant of a matrix is a scalar value that is a function of its entries. The determinant can offer insights into the properties of the matrix, such as whether it is invertible or how it can transform geometric objects. Evaluating the determinant becomes increasingly complex with larger matrices, which is why specific methods such as cofactor expansion are employed.

In our example, the matrix is given as:
\[\begin{bmatrix} 4 & 5 & -1 & -1 \2 & -3 & 1 & 0 \-5 & 1 & 3 & 9 \0 & -2 & 1 & 5 \end{bmatrix}\]
Each row and column consists of four elements, providing a total of sixteen elements. The challenge lies in connecting these elements to ascertain the matrix’s determinant. Calculating the determinant of a 4x4 matrix involves breaking it down into smaller parts using cofactor expansion.

Deciphering Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a technique used to compute the determinant of square matrices. This method involves expanding the determinant along a selected row or column while using cofactors. A cofactor is calculated by taking the determinant of a smaller matrix (submatrix) formed by deleting the row and column containing that element.

• Start by choosing a row or column for expansion. Preferably pick one with zeros for simpler calculations.
• Calculate the cofactor of each non-zero element in the chosen row or column.
• Use the formula: \( ext{det}(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + \ldots + a_{in}C_{in}\)
  where \(a_{ij}\) is an element of the matrix and \(C_{ij}\) is the cofactor of \(a_{ij}\).

This sequence of steps allows complex matrices to be broken down into simpler, more manageable sections for determinant calculations.

Exploring the 3x3 Submatrix

The concept of submatrices arises naturally in the computation of determinants through cofactor expansion. A 3x3 submatrix is formed by eliminating one row and one column from a 4x4 matrix. This reduced form makes it easier to calculate its determinant.

For instance, in our step-by-step solution, multiple 3x3 submatrices were used:
\[\begin{bmatrix}4 & 5 & -1 \2 & -3 & 1 \-5 & 1 & 3\end{bmatrix}, \\begin{bmatrix}4 & 5 & -1 \2 & -3 & 0 \-5 & 1 & 9\end{bmatrix}, \and \begin{bmatrix}4 & 5 & -1 \2 & -3 & 1 \-5 & 1 & 3\end{bmatrix}\]
Each submatrix needs its determinant calculated. The smaller 3x3 matrix's determinant is simpler since it uses basic determinant rules like: \[ ext{det}(A) = a_{11}(b_{22}b_{33} - b_{23}b_{32}) - a_{12}(b_{21}b_{33} - b_{23}b_{31}) + a_{13}(b_{21}b_{32} - b_{22}b_{31})\]
This calculation becomes fundamental in determining the overall 4x4 matrix determinant.

Grasping Matrix Algebra in Determinants

Matrix algebra is a specialized version of linear algebra dealing with matrix operations such as addition, multiplication, and finding determinants. Understanding matrix algebra is crucial for manipulating matrices and solving linear equations and systems.

In the context of determinant evaluation for a 4x4 matrix:

• The determinant is influenced by the matrix's elements and their arrangement.
• Cofactor expansion links matrix algebra with the practical application of calculating determinants.
• Each step, from identifying a row for expansion to calculating submatrix determinants, relies on principles of matrix algebra.

These principles ensure that even complex matrix operations like finding the determinant follow a structured approach grounded in mathematics. Embracing matrix algebra concepts allows students to handle matrices proficiently, be it for theoretical exploration or practical problem-solving.