Problem 8 - College Algebra with Modeling and Visualization

Question

Find the specified minor and cofactor for \(A\). $$ M_{31} \text { and } A_{31} \text { if } A=\left[\begin{array}{rrr} 0 & 0 & -1 \\ 6 & -7 & 1 \\ 8 & -9 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

The minor \(M_{31} = -7\) and the cofactor \(A_{31} = -7\).

Step 1: Identify Minor M31

The Minor, denoted as \(M_{31}\), is the determinant of the 2x2 matrix that remains after deleting the 3rd row and 1st column from matrix \(A\). Thus, matrix \(A\) becomes:\[\begin{bmatrix}0 & -1 \-7 & 1\end{bmatrix}\].

Step 2: Calculate Minor M31

Find the determinant of the matrix formed in Step 1:\[M_{31} = (0)(1) - (-1)(-7) = 0 - 7 = -7\].

Step 3: Identify Cofactor A31

The cofactor, \(A_{31}\), is calculated using the Minor \(M_{31}\) with the formula \(A_{ij} = (-1)^{i+j}M_{ij}\). Here, \(i = 3\) and \(j = 1\), so:\[A_{31} = (-1)^{3+1}M_{31} = (-1)^{4}(-7)\].

Step 4: Calculate Cofactor A31

Since \((-1)^4\) equals \(1\), the cofactor is:\[A_{31} = 1 \times (-7) = -7\].

Key Concepts

Minor of a Matrix

When working with matrices, understanding how to calculate the minor is an essential skill. The minor of a matrix refers to the determinant of a smaller square matrix derived from a larger one. This smaller matrix, often 2x2, is formed when you remove one row and one column from the original matrix.
In the original exercise above, we encountered a matrix \(A\) where we needed to find \(M_{31}\). This notation implies we are looking for the minor of the matrix at position 3rd row and 1st column.
By deleting this row and column from matrix \(A\), you obtain a submatrix from which the minor is calculated.

• Identify the row and column to remove. In our example, it is the third row and first column.
• Form the submatrix from the remaining elements.
• Calculate the determinant of this submatrix to find the minor. For the provided 2x2 matrix \(\begin{bmatrix}0 & -1\ -7 & 1\end{bmatrix}\), the minor \(M_{31}\) was determined to be \(-7\).

Understanding minors is crucial for calculating the cofactor and the determinant of larger matrices.

Cofactor

Once you have determined the minor of a matrix, the next step involves calculating the cofactor. Cofactors are integral in various matrix operations, including finding determinants. They include not just the minor, but also a sign change factor based on the position of the matrix element under consideration.
The formula for determining a cofactor \(A_{ij}\) is \((-1)^{i+j}M_{ij}\), combining the minor \(M_{ij}\) with a sign change that depends on the position \((i,j)\).

• Calculate the minor from the submatrix as explained earlier.
• Apply the sign factor. For instance, if \(i + j\) is even, the sign is positive; if odd, the sign is negative.
• In our example, \(A_{31} = (-1)^{3+1} \times M_{31} \), since 3+1 is even, the cofactor \(A_{31}\) is directly equal to \(-7\) as the sign factor is \(+1\).

Cofactors are utilized when expanding determinants along a row or column and offer insights into the properties and solutions of matrices.

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix. It is critical in many areas of linear algebra, including solving systems of linear equations, calculating the inverse of a matrix, and understanding the matrix’s properties. The process of finding a determinant involves understanding both the minors and cofactors of the matrix.
Determinants use a process called cofactor expansion along a row or column. This expansion involves taking each element of the row or column, finding its corresponding cofactor, and summing the results after multiplying by the respective elements.

• If you have a 3x3 matrix, choose any row or column for expansion.
• For each element chosen, calculate the minor and then the cofactor.
• Sum up all products of the original elements and their cofactors.

In our exercise, we only calculated one cofactor, but in determining the determinant of a full 3x3 matrix, you would perform this for each element in a chosen row or column. The determinant helps define if the matrix is singular or non-singular, influencing many results in algebraic solutions.