Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rr}11 & 6 \\\\-3 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

The minors are: [2, -3, 6, 11]. The cofactors are: [2, 3, -6, 11].

Step 1: Find the Minors

For a 2x2 matrix \[\left[\begin{array}{rr}a & b\\c & d\end{array}\right]\] the minors of \(a, b, c,\) and \(d\) are \(d, c, b,\) and \(a\) respectively. So for the given matrix \[\left[\begin{array}{rr}11 & 6\\-3 & 2\end{array}\right]\] the minors of \(11, 6, -3,\) and \(2\) are \(2, -3, 6,\) and \(11\) respectively.

Step 2: Find the Cofactors

The cofactor of an element in a matrix is the minor of that element multiplied by \((-1)^{(i+j)}\), where \(i\) is the row index and \(j\) is the column index (starting from index 1). For the given matrix, the cofactors of \(11, 6, -3,\) and \(2\) are \(2\times((-1)^{(1+1)}), -3\times((-1)^{(1+2)}), 6\times((-1)^{(2+1)}), 11\times((-1)^{(2+2)})\) which simplifies to \(2, 3, -6,\) and \(11\) respectively.

Key Concepts

Minors of a Matrix

When working with matrix algebra, especially when determining the inverse or determinant, understanding the concept of minors is essential. A minor of a matrix is derived from the determinant of a smaller matrix formed by excluding the row and column of a specific element.
A minor in a 2x2 matrix is rather straightforward. In fact, it simply refers to the single remaining value when you exclude the row and column of an element. For instance, given a matrix:

• If you focus on element 11, excluding the first row and column leaves you with 2, which is the minor of 11.
• Likewise, the minor of 6 is -3 since it is the only remaining element after excluding 6's row and column.

For our matrix:\[\left[\begin{array}{rr}11 & 6\-3 & 2\end{array}\right]\]the minors of elements 11, 6, -3, and 2 become 2, -3, 6, and 11 respectively.

Cofactors of a Matrix

Cofactors add a layer of complexity to the concept of minors by introducing a sign factor. This sign is determined by the position of the element within the matrix. The formula for calculating a cofactor involves the minor of the element and a sign, given by \((-1)^{(i+j)}\), where \(i\) and \(j\) represent the row and column indices respectively.
Here's how you calculate the cofactors:

• For the element 11 located at \((1,1)\), the cofactor is \(2 \times (-1)^{2} = 2\).
• For 6 at \((1,2)\), the cofactor is \(-3 \times (-1)^{3} = 3\).
• For -3 placed at \((2,1)\), the cofactor becomes \(6 \times (-1)^{3} = -6\).
• Finally, for 2 at \((2,2)\), it turns into \(11 \times (-1)^{4} = 11\).

This methodology transforms minors into cofactors, situating them precisely in the calculation of a matrix's determinant.

2x2 Matrix

A 2x2 matrix is the simplest form of a square matrix that is widespread in mathematics due to its basic structure and ease of computation. Its form is described as:
\[\left[\begin{array}{rr}a & b\c & d\end{array}\right]\]
This format allows for an array of computations, including determinants, inverses, and transformations.

To find the determinant of a 2x2 matrix, use:\[\text{Determinant} = ad - bc\]
This equation exemplifies why understanding minors (and subsequently cofactors) is vital, as these components contribute to determinant calculations in larger matrices. The determinant informs many matrix operations, influencing systems of equations, transformations, and more. The straightforward nature of a 2x2 matrix makes it a crucial learning stepping stone in matrix algebra.