Problem 17 - Algebra and Trigonometry

Question

Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{23}, A_{23} $$

Step-by-Step Solution

Answer

Both the minor \(M_{23}\) and the cofactor \(A_{23}\) are 0.

Step 1: Identify the Submatrix

To find the minor \(M_{23}\), we remove the 2nd row and 3rd column from matrix \(A\). This leaves us with the following 2x2 submatrix: \[\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix} \]

Step 2: Calculate the Determinant of the Submatrix

The minor \(M_{23}\) is the determinant of the submatrix obtained in Step 1. For the submatrix \(\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\), the determinant is calculated as follows: \[(1 \times 0) - (0 \times 0) = 0 \] Thus, \(M_{23} = 0\).

Step 3: Calculate the Cofactor

The cofactor \(A_{23}\) is calculated using the formula \(A_{ij} = (-1)^{i+j} M_{ij}\). Here, \(i=2\) and \(j=3\), so \[A_{23} = (-1)^{2+3} M_{23} = (-1)^5 \times 0 = 0 \.\]

Step 4: Conclusion: Solution Summary

The minor \(M_{23}\) is 0, and the cofactor \(A_{23}\) is also 0.

Key Concepts

Minors and Cofactors

In matrix algebra, minors and cofactors are essential concepts for calculating determinants and understanding matrix properties. A minor, denoted as \(M_{ij}\), of an element in a matrix, is the determinant of the submatrix formed by deleting the \(i\)-th row and \(j\)-th column of an original matrix. It helps isolate specific parts of a matrix for evaluation purposes. For instance, in the provided matrix \(A\), the minor \(M_{23}\) is computed by removing the second row and third column, resulting in a 2x2 submatrix \(\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\).
The concept of cofactors comes into play when you're working with these minors. The cofactor, represented as \(A_{ij}\), is calculated using the minor and a sign factor, \((-1)^{i+j}\). This factor alternates between positive and negative depending on the sum of the indices \(i\) and \(j\). For the same example in matrix \(A\), the cofactor \(A_{23}\) is computed using the formula:

• \(A_{23} = (-1)^{2+3} M_{23} = (-1)^{5} \times 0 = 0\)

This process highlights how cofactors are not only dictated by the size of the submatrix but also by their position in the original matrix.
Understanding these concepts is crucial because they form the building blocks for more complex calculations, such as determining the inverse of a matrix and calculating determinants.

Determinant Calculation

The determinant is a special numerical value that is calculated from a square matrix. It provides important properties of the matrix, such as whether it's invertible. Determinants are computed using specific rules depending on the size of the matrix, making the understanding of minors and cofactors quite relevant.
To calculate the determinant of a 2x2 matrix like \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), use the formula:

• Determinant = \(ad - bc\)

In this example, when calculating the determinant of the submatrix \(\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\) from the original 3x3 matrix \(A\), we apply the formula:

• Determinant = \((1 \times 0) - (0 \times 0) = 0\)

Determinants of higher order matrices, such as 3x3 and above, involve expanding along a row or column using minors and cofactors.
In conclusion, understanding how determinants link together with minors and cofactors demonstrates their significance in matrix algebra. They help unlock more complex matrix-related problems and are indispensable in various scientific and engineering calculations.

Step by Step Matrix Evaluation

Evaluating a matrix involves a systematic approach to solve or simplify problems involving the matrix. Whether you're determining a minor, calculating a determinant, or finding cofactors, a step-by-step method ensures accuracy and understanding.
The first step in a matrix evaluation often involves identifying the required components or submatrices, just like removing rows and columns to find minors. In our example, we removed the 2nd row and 3rd column to isolate the 2x2 submatrix, simplifying our problem to calculating the minor.
Once the submatrix is identified, the next step is determinant calculation, which we've examined closely. The minor \(M_{23}\) gives us 0 since the calculation \((1 \times 0) - (0 \times 0)\) resulted in zero.
The final step involves applying this minor in a broader context, such as calculating a cofactor. Here, the formula \(A_{ij} = (-1)^{i+j} M_{ij}\) is used to introduce the sign modifications from the cofactor's perspectives.
Simply put:

• Identify the submatrix.
• Compute the determinant for the minor.
• Calculate the cofactor using the minor and sign adjustment.

By closely following these steps, matrix evaluation becomes much more manageable and straightforward, especially when learners approach matrix algebra problems with these structured strategies.