Problem 10 - A Graphical Approach to Precalculus with Limits

Question

For each element in the second row of the given matrix, find its cofactor (See Example 3 .) $$\left[\begin{array}{rrr}1 & -1 & 2 \\\1 & 0 & 2 \\\0 & -3 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

The cofactors are -5, 1, and 3.

Step 1: Identify the Matrix

The given matrix is a 3x3 matrix: \(\begin{bmatrix}1 & -1 & 2 \1 & 0 & 2 \0 & -3 & 1\end{bmatrix} \). We will find cofactors for each element in the second row, which are 1, 0, and 2.

Step 2: Calculate Cofactor of Element (2,1)

To find the cofactor of the element at position (2,1), which is 1, remove the second row and first column to get the submatrix: \(\begin{bmatrix}-1 & 2 \-3 & 1\end{bmatrix}\). Calculate its determinant: \((-1)(1) - (2)(-3) = -1 + 6 = 5\). Apply the sign based on the position, which is \((-1)^{2+1} = -1\). The cofactor is \(-1 imes 5 = -5\).

Step 3: Calculate Cofactor of Element (2,2)

For the element at position (2,2), which is 0, remove the second row and second column to get the submatrix: \(\begin{bmatrix}1 & 2 \0 & 1\end{bmatrix}\). Determine its determinant: \((1)(1) - (2)(0) = 1\). The sign factor is \((-1)^{2+2} = 1\). Hence, the cofactor is \(1 \times 1 = 1\).

Step 4: Calculate Cofactor of Element (2,3)

For the element at position (2,3), which is 2, remove the second row and third column to form the submatrix: \(\begin{bmatrix}1 & -1 \0 & -3\end{bmatrix}\). Calculate its determinant: \((1)(-3) - (-1)(0) = -3\). The sign factor is \((-1)^{2+3} = -1\). Thus, the cofactor is \(-1 \times -3 = 3\).

Key Concepts

Determinant Calculation

Calculating the determinant of a matrix is a key mathematical process. This process helps determine whether a matrix has an inverse, among other properties. To compute the determinant of a 2x2 submatrix, apply the form: \[|A| = ad - bc\]where \(a\), \(b\), \(c\), and \(d\) are the elements of the submatrix. For example, in the submatrix \(\begin{bmatrix}-1 & 2 \ -3 & 1\end{bmatrix}\), - The calculation is \((-1) \times 1 - 2 \times (-3) = -1 + 6 = 5\).This determinant calculation is the crux of determining cofactor values. Remember the specific position affects the result due to alternating signs. Understanding this is crucial in matrix operations.

3x3 Matrix

The 3x3 matrix is a standard square matrix often used in educational exercises. It consists of three rows and three columns. This matrix is sometimes called a second-order square matrix. In our exercise, the given matrix is:\[\begin{bmatrix}1 & -1 & 2 \ 1 & 0 & 2 \ 0 & -3 & 1\end{bmatrix}\]Working with a 3x3 matrix involves specific techniques, especially when finding cofactors. Each element's cofactor in a 3x3 matrix is found by isolating smaller 2x2 matrices.- This involves excluding the relevant row and column of the targeted element.Being comfortable with navigating these smaller matrices is key in matrix computations.

Submatrix Extraction

Submatrix extraction is the process of isolating a smaller matrix from a larger one. This is crucial for finding cofactors in larger matrices. When extracting a submatrix, temporarily remove the row and column of the element in question. For example, to find the cofactor for position (2,1) of the given 3x3 matrix:- Remove the second row and first column.- The resulting submatrix is \(\begin{bmatrix}-1 & 2 \ -3 & 1\end{bmatrix}\).This smaller matrix then allows you to calculate a determinant. Submatrix extraction is foundational for solving many matrix algebra problems efficiently. Mastering this concept streamlines processes like cofactor expansions or inverse calculations.