Problem 20 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rrr}-2 & 9 & 4 \\\7 & -6 & 0 \\\6 & 7 & -6\end{array}\right]$$

Step-by-Step Solution

Answer

The computed minors and cofactors for the matrix are respectively: minors = [[36, -42, 13], [-244, -24, -68], [36, 8, -51]] and cofactors = [[36, 42, -13], [244, -24, 68], [-36, -8, -51]].

Step 1: Compute the minors

We start by calculating the minors for each element in the matrix:The minor of the element -2 is M_{11} = det([[-6, 0], [7, -6]]) = -6*(-6) - 0*7 = 36;The minor of the element 9 is M_{12} = det([[7, 0], [6, -6]]) = 7*(-6) - 0*6 = -42;The minor of the element 4 is M_{13} = det([[7, -6], [6, 7]]) = 7*7 - 6*6 = 49 -36 =13;The minor of the element 7 is M_{21} = det([[9, 4], [7, -6]]) =-216 -28 = -244;The minor of the element -6 is M_{22} = det([[-2, 4], [6, -6]]) =(-2)*(-6) - 4*6 =-24;The minor of the element 0 is M_{23} = det([[-2, 9], [6, 7]]) = -2*7 - 6*9 = -14 -54 = -68;The minor of the element 6 is M_{31} = det([[9, 4], [-6, 0]]) = 0 + 36 =36;The minor of the element 7 is M_{32} = det([[-2, 4], [7, 0]]) = 0+8 = 8;The minor of the element -6 is M_{33} = det([[-2, 9],[7, -6]]) = 12 -63 = -51.

Step 2: Compute the cofactors

Now, time to calculate the cofactors. The cofactor is given by C_{ij} = (-1)^(i+j) * M_{ij}:The cofactor of the element -2 is C_{11} = (-1)^(1+1)* M_{11} = 36;The cofactor of the element 9 is C_{12} = (-1)^(1+2) * M_{12} = 42;The cofactor of the element 4 is C_{13} = (-1)^(1+3) * M_{13} = -13;The cofactor of the element 7 is C_{21} = (-1)^(2+1) * M_{21} = 244;The cofactor of the element -6 is C_{22} = (-1)^(2+2) * M_{22} = -24;The cofactor of the element 0 is C_{23} = (-1)^(2+3) * M_{23} = 68;The cofactor of the element 6 is C_{31} = (-1)^(3+1) * M_{31} = -36;The cofactor of the element 7 is C_{32} = (-1)^(3+2) * M_{32} = -8;The cofactor of the element -6 is C_{33} = (-1)^(3+3) * M_{33} = -51.

Key Concepts

Minors

Understanding minors in a matrix helps unlock the secrets behind matrix determinants. A "minor" of an element in a 3x3 matrix is the determinant of the 2x2 matrix that remains after removing the row and column containing that element. It plays a crucial role in matrix algebra by enabling calculations of eigenvalues, inverses, and determinants.

• To find a minor, choose an element in the matrix, such as the element on the first row and first column, \(-2\).
• Next, eliminate the row and column that intersect at that element.
• Then, compute the determinant of the smaller 2x2 matrix left over. For example, for the element \(-2\), the associated 2x2 matrix is \[ \begin{bmatrix} -6 & 0 \ 7 & -6 \ \end{bmatrix}\]. The minor is then found by calculating \(M_{11} = (-6 \times -6) - (0 \times 7) = 36\).

By finding the minors for each element, one begins to unravel the determinants in a larger and more complex matrix.

Cofactors

Cofactors sound similar to minors, but there's a slight twist in the process. Cofactors are modifications of minors with a sign change. Calculated from minors using the formula \((-1)^{i+j} \times M_{ij}\), where \(i\) and \(j\) are the row and column indices of the matrix element. This modification helps embrace the alternating sign pattern essential for calculating determinants and other matrix operations.

• Start by finding the minor of an element, like you did with the 3x3 matrix. For instance, with the minor \(M_{11} = 36\), locate its corresponding row and column summation: \(1 + 1 = 2\).
• Apply the cofactor formula: multiply the minor by \((-1)^2\) to get the cofactor \(C_{11} = 36\).
• Remember that these signs alternate: for example, for the second row, first column \(C_{21} = (-1)^{2+1} \times M_{21}\).

This concept equips you with a powerful mathematical tool, useful in diverse calculations, such as evaluating determinants of larger matrices.

3x3 Matrix

A 3x3 matrix is a compact grid of numbers organized into three rows and three columns. It forms a basis for solving a range of mathematical problems from systems of equations to transformations in geometry. Grasping the fundamental traits of a 3x3 matrix directly aids in understanding broader matrix-related concepts.

• This type of matrix holds nine elements, each playing a role in larger calculations. Operations can include addition, subtraction, and multiplication.
• However, one key operation unique to square matrices like our 3x3 is the determinant. This critical value can, among other things, indicate the invertibility of a matrix.
• By processing each element through calculating minors, and then cofactors, you create a panoramic understanding of determinants from top to bottom of the matrix.

Within the foundational structure of a 3x3 matrix, rich mathematical analysis unfolds, empowering your journey through more advanced algebraic principles.