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Chapter 9: Problem 12

Find the cofactor of each element in the second row for each determinant. $$\left|\begin{array}{rrr} 1 & -1 & 2 \\ 1 & 0 & 2 \\ 0 & -3 & 1 \end{array}\right|$$

Short Answer

Expert verified
The cofactors of the elements in the second row are -5, 1, and 3, respectively.

Step by step solution

01

Identify the element

Focus on the second row of the determinant: 1, 0, 2.
02

Calculate cofactor for element (2,1)

To find the cofactor of the element at (2,1), ignore the row and column of the element. The resulting minor determinant is: $$\begin{vmatrix} -1 & 2 \ -3 & 1 \ \text{Cofactor} = (-1)^{2+1} \times \begin{vmatrix} -1 & 2 \ -3 & 1 \ \text{Value} = (-1)^{3} \times \big[(-1)(1) - (2)(-3)\big]$$ = $$(-1) \times (-1 + 6) = -5$$.
03

Calculate cofactor for element (2,2)

To find the cofactor of the element at (2,2), ignore the row and column of the element. The resulting minor determinant is: $$\begin{vmatrix} 1 & 2 \ 0 & 1 \ \text{Cofactor} = (-1)^{2+2} \times \begin{vmatrix} 1 & 2 \ 0 & 1 \ \text{Value} = 1 \times \big[(1)(1) - (2)(0)\big] = 1$$.
04

Calculate cofactor for element (2,3)

To find the cofactor of the element at (2,3), ignore the row and column of the element. The resulting minor determinant is: $$\begin{vmatrix} 1 & -1 \ 0 & -3 \ \text{Cofactor} = (-1)^{2+3} \times \begin{vmatrix} 1 & -1 \ 0 & -3 \ \text{Value} = (-1) \times \big[(1)(-3) - (-1)(0)\big] = (-1) \times -3 = 3$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Cofactor
In matrix algebra, the cofactor of an element is a crucial concept. The cofactor of an element in a matrix is found by removing the row and column containing that element and taking the determinant of the remaining submatrix, called the minor determinant. Then, we multiply this minor determinant by \((-1)^{i+j}\), where \((i,j)\) are the coordinates of the element.

The sign factor \((-1)^{i+j}\) ensures the correct sign for the cofactor. If the sum of the row and column indices is even, the sign will be positive; if odd, the sign will be negative.
Finding the Minor Determinant
To find the minor determinant of an element in a matrix involves creating a smaller matrix. This smaller matrix is generated by removing the element's row and column from the original matrix.

For instance, if you have a matrix element at position (2,1), you ignore the second row and the first column. What鈥檚 left is a 2x2 submatrix. The determinant of this submatrix is calculated and used as part of the cofactor.

Knowing how to calculate the minor determinant quickly is essential for solving larger matrices and finding cofactors.
Determinant Calculation
The determinant calculation of a 2x2 matrix \(\begin{bmatrix}a & b \ c & d\buildmatrix}\) is straightforward. The formula is \(ad - bc\). Applying this formula to our extracted minor determinants simplifies finding the cofactor.

This practice is fundamental in matrix algebra and helps in determining properties like the invertibility of matrices. In our example, the determinant calculation for element (2,1) resulted in \(鈥�5\), demonstrating this crucial step in the cofactor computation.
Matrix Algebra Applications
Matrix algebra extends beyond just calculating determinants and cofactors. It forms the backbone of various fields like physics, computer science, and engineering. Understanding how to manipulate matrices enables solving complex systems of equations and transforming space through linear transformations.

Whether calculating the inverse of a matrix or solving quadratic forms, matrix algebra serves as a powerful tool. The understanding of finding cofactors and minor determinants feeds directly into more advanced topics, including eigenvalues and eigenvectors. Appreciating these foundational steps is vital for mastering higher-level concepts in mathematics and related disciplines.

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