Problem 40 - Precalculus with Limits

Question

In Exercises 39-54, find the determinant of the matrix.Expand by cofactors on the row or column that appears to make the computations easiest. \(\left[ \begin{array}{r} -2 & 2 & 3 \\ 1 & -1 & 0 \\ 0 & 1 & 4 \end{array} \right]\)

Step-by-Step Solution

Answer

The determinant of the matrix is -3.

Step 1: Identify the matrix

Identify the 3x3 matrix for which the determinant needs to be found. The matrix is \[\begin{bmatrix}-2 & 2 & 3 \ 1 & -1 & 0 \ 0 & 1 & 4\end{bmatrix}\]

Step 2: Choose the row or column to expand

Choose the row or column that will make computation easiest. Expand by cofactors on the 3rd column because of the 0 which makes computation easier.

Step 3: Compute the determinant

The determinant of a 3x3 matrix A can be calculated using the formula:\[det(A) = a_{ij}C_{ij} + a_{ij}C_{ij} + a_{ij}C_{ij}\]where \(a_{ij}\) are the elements of the chosen row or column and \(C_{ij}\) are the corresponding cofactors. Therefore, starting with the 3rd column, the determinant is calculated as follows:\[det(A) = 3*(-1)^{1+3}det([1,-1; 0,1]) - 0*(-1)^{2+3}det([-2,2; 0,1]) + 4*(-1)^{3+3}det([-2,2; 1,-1]) \]Computing each of the 2x2 determinants:\[det([1,-1; 0,1]) = (1)(1) - (0)(-1) = 1\]\[det([-2,2; 0,1]) = (-2)(1) - (0)(2) = -2\]\[det([-2,2; 1,-1]) = (-2)(-1) - (1)(2) = 0\]Therefore, the final determinant is:\[det(A) = 3*(-1)*(1) - 0*(1)*(-2) + 4*(1)*(0) = -3\]

Key Concepts

Cofactor Expansion

To find the determinant of a matrix, you can use a method called cofactor expansion. It is particularly useful when dealing with larger matrices. Cofactor expansion involves expressing a determinant as a combination of determinants of smaller matrices, known as minors. This method allows you to break down complex calculations and focus on simpler elements within the matrix.

In cofactor expansion, each element of the matrix has a corresponding cofactor, which is essentially the determinant of a smaller matrix obtained by removing the row and column of the element. The formula for the determinant of a matrix using cofactor expansion is:
\[ \text{det}(A) = a_{ij}C_{ij} \]

Here, \(a_{ij}\) represents the element and \(C_{ij}\) its cofactor.

• Cofactor is computed using: \((-1)^{i+j} \times \text{det of minor}(M)\)
• Minor is the smaller matrix found by removing the row and column of the element

This method is flexible, allowing you to choose any row or column for expansion, usually the one that simplifies the computation the most.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. The structure of such a matrix permits various operations, one of which is finding its determinant. The determinant of a 3x3 matrix is a special number that provides useful insights into the matrix, such as whether it is invertible.

In a 3x3 matrix, like the one given in the exercise \( \begin{bmatrix} -2 & 2 & 3 \ 1 & -1 & 0 \ 0 & 1 & 4 \end{bmatrix} \), each element plays a role in its determinant. The process involves:

• Choosing a row or column to perform cofactor expansion.
• Calculating minors by excluding row and column of a chosen element.
• Multiplying each element by its corresponding cofactor.
• Summing these products to find the determinant.

These matrices are central to many applications such as systems of linear equations and transformations.

Linear Algebra

Linear algebra is a branch of mathematics that studies vectors, matrices, and linear transformations. It is foundational for understanding complex mathematical structures and has applications in various fields such as engineering, physics, computer science, and more.

Determinants, like those calculated in the exercise, are an important concept in linear algebra. They help determine certain properties of matrices, including:

• Whether a matrix is invertible (a non-zero determinant indicates invertibility).
• Characteristics of a linear system of equations (such as unique solutions, infinite solutions, or no solution).
• Eigenvalues and eigenvectors.

In the context of the exercise, finding determinants can reveal insights about matrix transformations, stability of systems, and consistency of vector spaces. Understanding these concepts is critical for anyone dealing with linear transformations or systems in higher mathematics.