Question

In Exercises \(71-74\), find the determinant of the matrix. Tell which method you used. $$ \left[\begin{array}{rrr} 2 & 1 & 3 \\ 7 & 3 & -2 \\ 4 & 1 & 1 \end{array}\right] $$

Short answer

The determinant of the given matrix is -10. The method used is cofactor expansion along the first row.

Step-by-step solution

Preparation

The original matrix is \[ \left[\begin{array}{rrr} 2 & 1 & 3 \\ 7 & 3 & -2 \\ 4 & 1 & 1 \end{array}\right] \]. In order to use the method of co-factor expansion along the first row, prepare the matrix by identifying the elements of the first row, which are 2, 1, and 3.

Create submatrices

For each element of the first row, form 2x2 submatrices. To do this, exclude the row and column that each element is in. The submatrix for element 2 is \[\left[\begin{array}{rr} 3 & -2 \\ 1 & 1 \end{array}\right]\]. The submatrix for element 1 is \[\left[\begin{array}{rr} 7 & -2 \\ 4 & 1 \end{array}\right]\]. The submatrix for 3 is \[\left[\begin{array}{rr} 7 & 3 \\ 4 & 1 \end{array}\right]\].

Calculate the determinant of submatrices

The determinant of a 2x2 matrix \[\left[\begin{array}{rr} a & b \\ c & d \end{array}\right]\] is given by \(ad-bc\). Follow the same rule to calculate the determinant of each submatrix. The determinant of submatrix related with element 2 is \(3*1 - 1*-2 = 5\). The determinant of submatrix for element 1 is \(7*1 - 4*-2 = 15\). The determinant of submatrix for element 3 is \(7*1 - 3*4 = -5\).

Apply cofactor expansion

Now apply the co-factor expansion to find the total determinant which involves multiplying each element of the first row by the determinant of its corresponding submatrix and the sign based on its position (+ for even positions and - for odd ones). Here, the determinant is given by \(D = 2*5 - 1*15 + 3*(-5) = -10\).

Key concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a matrix. It's particularly useful for larger matrices where direct calculation would be cumbersome. Here's how it works:

• Choose a row or a column of the matrix to expand along, typically one with zeros for simplicity, though it’s often the first row or column in educational settings.
• For each element in the selected row or column, calculate its associated cofactor. The cofactor is determined by multiplying the determinant of a smaller matrix—formed by removing the row and column of the element—by either +1 or -1. The sign is determined by the position of the element: "+" for positions with coordinates' sum being even, and "-" for odd.
• Multiply each element of the row/column by its corresponding cofactor, then sum these products. This total is the matrix's determinant.

Understanding cofactor expansion is crucial because it not only helps in finding determinants but also lays the foundation for other operations like finding inverses and solving systems of linear equations.

Submatrices

Submatrices are smaller matrices extracted from a larger matrix by removing specific rows and columns. When performing cofactor expansion, submatrices become essential as they help in evaluating determinants of larger matrices.

• To form a submatrix for cofactor expansion, choose an element in the matrix and eliminate its row and column.
• The resulting submatrix allows you to focus on smaller data sets, making complex calculations more manageable.
• In our exercise, the original 3x3 matrix was broken into three 2x2 submatrices, each corresponding to a term in the first row.

The role of submatrices extends beyond just determinants; they also appear in various transformations and operations in linear algebra, making them a fundamental concept to grasp.

2x2 Determinant Calculation

Calculating the determinant of a 2x2 matrix is considerably simpler than that of larger matrices, and it is crucial in cofactor expansion and other operations. For any 2x2 matrix:\[\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\]the determinant is computed as \( ad - bc \).

• This formula stems from the geometric interpretation of determinants, representing an area or a volume in multi-dimensional spaces.
• It is vital to apply this correctly as it becomes the base for larger determinants via cofactor expansion.
• In our specific exercise, each of the submatrices’ determinants was computed using this formula, involving simple arithmetic calculations.

A solid grasp of this calculation helps in understanding and performing more complex operations in linear algebra!

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, matrices, vector spaces, and more. Determinants, including those calculated using cofactor expansions, form an integral component of linear algebra.

• Determinants can provide insights into the properties of a matrix, such as whether it is invertible or singular.
• Understanding how to compute a determinant is foundational for many topics, including eigenvalues, matrix inverses, and systems of linear equations.
• Beyond theory, linear algebra has practical applications across numerous fields, from computer science to engineering and data analysis.
• For students, mastering basic concepts like matrix operations and determinants paves the way for more advanced studies in the subject.

Therefore, understanding the basics such as cofactor expansion and submatrices is essential for anyone delving deeper into linear algebra.