Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rr}3 & 4 \\\2 & -5\end{array}\right]$$

Step-by-Step Solution

Answer

The minors of the given matrix are -5, 2, 3, 4. The cofactors are 5, -2, -3, 4.

Step 1: Find the Minors of the Matrix

To find minors for 2x2 matrix, get the determinant of the 1x1 matrix formed by removing row and column for each element. In other words, minor of any element at position \(i,j\) for a 2x2 matrix is the single term left after removing the \(i\)th row, and \(j\)th column. So, the minors are: Minor(3)=(-5), Minor(4)=2, Minor(2)=3, Minor(-5)=4.

Step 2: Find the Cofactors of the Matrix

Cofactor of an element is obtained by multiplying its minor with \((-1)^{i+j}\) where \(i\) and \(j\) are the row and column number of the element. So the cofactors are: Cofactor(3)= -5(-1)^(1+1)= 5, Cofactor(4)=2(-1)^(1+2)= -2, Cofactor(2)=3(-1)^(2+1)=-3, Cofactor(-5)=4(-1)^(2+2)=4.

Key Concepts

Matrix Minors

When dealing with matrices, especially when calculating determinants or applying cofactor expansion, we often hear about the term "minors." Simply put, a minor of a matrix is the determinant of a smaller matrix created by removing one row and one column from the original matrix. In the case of a 2x2 matrix, finding the minor is relatively simple. For each element, we remove its respective row and column, leaving us with a single number, which effectively becomes the minor of that element. For example, given the 2x2 matrix \[\begin{bmatrix} 3 & 4 \ 2 & -5 \end{bmatrix}\]- For element 3, remove the first row and first column, leaving -5 as the minor.- For element 4, remove the first row and second column, leaving 2 as the minor.- For element 2, remove the second row and first column, leaving 3 as the minor.- For element -5, remove the second row and second column, leaving 4 as the minor.This concept of minors becomes particularly useful for more complex matrices when calculating determinants through cofactor expansion.

Cofactor Expansion

The concept of cofactor expansion is crucial when calculating the determinant of a matrix. A cofactor is derived from the minor of a matrix element, but it also accounts for the position of the element by introducing a sign change. Essentially, a cofactor is the minor multiplied by \((-1)^{i+j}\) , where \(i\) and \(j\) are the respective row and column indices of the element. For the 2x2 matrix \[\begin{bmatrix} 3 & 4 \ 2 & -5 \end{bmatrix}\]Cofactor calculations are as follows:

• The cofactor of 3 is 5, calculated as \(-5(-1)^{1+1} = 5\).
• The cofactor of 4 is -2, calculated as \(2(-1)^{1+2} = -2\).
• The cofactor of 2 is -3, calculated as \(3(-1)^{2+1} = -3\).
• The cofactor of -5 is 4, calculated as \(4(-1)^{2+2} = 4\).

The series of cofactors can then be used to find the determinant of a larger matrix if needed, with each cofactor contributing a signed value to the sum.

2x2 Matrix

Understanding a 2x2 matrix is foundational in linear algebra, offering a stepping stone to more complex matrix operations. A 2x2 matrix consists of two rows and two columns. Its general form can be denoted as:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Here, \(a, b, c,\) and \(d\) are the elements of the matrix. The calculation of the determinant, minors, and cofactors is simpler for a 2x2 matrix than a larger one. Working with 2x2 matrices provides an excellent introduction to these processes while still being manageable for thorough understanding. In essence, mastering 2x2 matrices gives students the fundamental skills needed to tackle more complex scenarios.

Determinants Calculation

Calculating the determinant of a matrix is a key operation in linear algebra with numerous applications, such as solving systems of equations and understanding matrix invertibility. For a 2x2 matrix, the determinant is calculated directly using the formula:\[ad - bc\]where the matrix is:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]Thus, for the matrix\[\begin{bmatrix} 3 & 4 \ 2 & -5 \end{bmatrix}\]The determinant is:\[3(-5) - 4(2) = -15 - 8 = -23\]The determinants help determine the uniqueness of solutions to systems of linear equations. A non-zero determinant indicates a unique solution, while a zero determinant suggests no or infinite solutions. Determinants also give insights into the geometric transformations that a matrix represents, such as rotation and scaling, in a geometric space.