Problem 12 - Algebra and Trigonometry

Question

\(9-14\) Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{13}, A_{13} $$

Step-by-Step Solution

Answer

The minor \( M_{13} \) is 0, and the cofactor \( A_{13} \) is also 0.

Step 1: Identify the submatrix for the Minor

To find the minor \( M_{13} \), we need to delete the first row and the third column from matrix \( A \). This leaves us with the submatrix: \[ \begin{bmatrix} -3 & 5 \ 0 & 0 \end{bmatrix} \]

Step 2: Calculate the Determinant of the Submatrix

The determinant of the submatrix \( \begin{bmatrix} -3 & 5 \ 0 & 0 \end{bmatrix} \) is calculated by using the formula for a \( 2 \times 2 \) matrix: \( ad - bc \). Here, \( a = -3, b = 5, c = 0, d = 0 \). Therefore, the determinant is \( (-3) \times 0 - 5 \times 0 = 0 \).

Step 3: State the Minor

The minor \( M_{13} \) is the determinant of the submatrix, which is \( 0 \).

Step 4: Calculate the Cofactor

The cofactor \( A_{13} \) is calculated by multiplying the minor \( M_{13} \) by \( (-1)^{1+3} \). Since \( M_{13} = 0 \) and \( (-1)^{4} = 1 \), we find that \( A_{13} = 0 \times 1 = 0 \).

Step 5: State the Cofactor

The cofactor \( A_{13} \) for the element \( a_{13} \) is \( 0 \).

Key Concepts

Matrix Minors

Matrix minors are a crucial concept in linear algebra. A minor of a matrix is the determinant of a specific submatrix formed by removing one row and one column from the original matrix. Each element in a matrix has an associated minor. To find a minor, choose the element for which you want to calculate the minor, then delete its row and column. For instance, with our given matrix \(A\), to calculate the minor \(M_{13}\), we remove the first row and the third column. This leaves us with the submatrix \[\begin{bmatrix}-3 & 5 \0 & 0\end{bmatrix}\]Calculating the determinant of this submatrix gives us the minor for the element at position \((1,3)\). The determinant is found using the formula for a \(2 \times 2\) matrix: \(ad - bc\). Here the values are \(a = -3\), \(b = 5\), \(c = 0\), and \(d = 0\), leading to a determinant of \(0\). Thus, the minor \(M_{13}\) is \(0\).

Matrix Cofactors

In addition to minors, the concept of cofactors is also essential. A cofactor is similar to a minor but includes a sign change based on the position of the element in the matrix. Specifically, the cofactor associated with an element \(a_{ij}\) is calculated by multiplying its minor \(M_{ij}\) by \((-1)^{i+j}\). This sign comes from a checkerboard pattern where elements in positions where \(i + j\) is even retain their sign, while elements in positions with odd \(i + j\) are negated. Using our matrix \(A\), the cofactor \(A_{13}\) for element \(a_{13}\) can be found by using its minor \(M_{13} = 0\). The sign factor is \((-1)^{1+3} = 1\), as the sum \(1+3 = 4\) is even. Therefore, \(A_{13} = 0 \times 1 = 0\). The cofactor here is \(0\). The process of calculating cofactors is fundamental in computing matrix determinants of larger matrices.

2x2 Submatrices

Understanding \(2 \times 2\) submatrices is vital, especially since they are at the heart of calculating minors and determinants. When dealing with larger matrices, a \(2 \times 2\) submatrix is a smaller square matrix that we often use to simplify these calculations.To extract a \(2 \times 2\) submatrix from a larger matrix like matrix \(A\), you remove one row and one column. For finding the minor \(M_{13}\), we deleted the first row and third column, resulting in the submatrix \[\begin{bmatrix}-3 & 5 \0 & 0\end{bmatrix}\]The determinant of this submatrix is critical as it influences the minor and subsequently the cofactor for elements of the larger matrix.Remember, the determinant of any \(2 \times 2\) matrix \(\begin{bmatrix}a & b \ c & d \end{bmatrix}\) is calculated as \(ad - bc\). This simple operation is foundational in linear algebra, and mastering it will facilitate better understanding and manipulation of more complex matrices.