Problem 1 - Precalculus: Functions and Graphs

Question

Find all the minors and cofactors of the elements in the matrix. $$\left[\begin{array}{rr} 7 & -1 \\ 5 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

Minors: [0, 5, -1, 7]; Cofactors: [0, -5, 1, 7].

Step 1: Identify the Matrix Elements

The given matrix is a 2x2 matrix: \[A = \begin{bmatrix} 7 & -1 \ 5 & 0 \end{bmatrix}\]The elements of the matrix are: - \( a_{11} = 7 \)- \( a_{12} = -1 \)- \( a_{21} = 5 \)- \( a_{22} = 0 \).

Step 2: Calculate Minors of the Matrix

The minor of an element \( a_{ij} \) in a 2x2 matrix is the determinant of the 1x1 matrix obtained by deleting its row and column.- Minor of \( a_{11} = 7 \): Remove the first row and first column. Minor = determinant of \( [0] \) = 0.- Minor of \( a_{12} = -1 \): Remove the first row and second column. Minor = determinant of \( [5] \) = 5.- Minor of \( a_{21} = 5 \): Remove the second row and first column. Minor = determinant of \( [-1] \) = -1.- Minor of \( a_{22} = 0 \): Remove the second row and second column. Minor = determinant of \( [7] \) = 7.

Step 3: Calculate Cofactors of the Matrix Elements

The cofactor of an element \( a_{ij} \) is given by the formula \( C_{ij} = (-1)^{i+j} \times \text{Minor of } a_{ij} \).- Cofactor of \( a_{11} = 7 \): \( C_{11} = (-1)^{1+1} \times 0 = 0 \).- Cofactor of \( a_{12} = -1 \): \( C_{12} = (-1)^{1+2} \times 5 = -5 \).- Cofactor of \( a_{21} = 5 \): \( C_{21} = (-1)^{2+1} \times (-1) = 1 \).- Cofactor of \( a_{22} = 0 \): \( C_{22} = (-1)^{2+2} \times 7 = 7 \).

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special value that can be calculated from a square matrix. It is a scalar attribute that provides crucial information about the matrix, such as whether it is invertible or, in geometric terms, the scaling factor of the space described by the matrix. For a 2x2 matrix like \( A = \begin{bmatrix} 7 & -1 \ 5 & 0 \end{bmatrix} \), the determinant can be calculated using a simple formula: \[\text{det}(A) = a_{11} \times a_{22} - a_{12} \times a_{21}. \]In this exercise, substitute the elements from the matrix into the formula:- \( a_{11} = 7 \)- \( a_{12} = -1 \)- \( a_{21} = 5 \)- \( a_{22} = 0 \)So, the determinant is:\[\text{det}(A) = 7 \times 0 - (-1) \times 5 = 0 + 5 = 5.\]This tells us that the matrix is invertible and does not collapse into a lower-dimensional space. Understanding the determinant is crucial for grasping broader concepts like matrix inversion and eigenvalues.

2x2 Matrix

A 2x2 matrix, as the name suggests, consists of two rows and two columns. It is one of the simplest forms of matrices, yet quite rich in application.Consider a general form of a 2x2 matrix:\[B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}.\]Each position in the matrix is identified by the row and column it occupies:- The element \( b_{11} \) is in the first row, first column.- The element \( b_{12} \) is in the first row, second column.- The element \( b_{21} \) is in the second row, first column.- The element \( b_{22} \) is in the second row, second column.In the exercise, identifying these positions helps in the calculation of minors and cofactors, which simplifies many matrix operations.2x2 matrices play significant roles in systems of linear equations, transformations, and rotations in 2D space. Despite their simplicity, mastering their characteristics lays down a strong foundation for understanding higher-dimensional matrices.

Matrix Elements

Matrix elements are the individual numbers or symbols that make up a matrix. Each element is crucial as it contributes to various operations such as addition, multiplication, or finding determinants and inverses.For a 2x2 matrix:\[C = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix},\]each element can be denoted by its row and column number:- \( c_{11} \) is the element in the first row and first column.- \( c_{12} \) is in the first row, second column.- \( c_{21} \) is in the second row, first column.- \( c_{22} \) is in the second row, second column.In context with the given matrix from the exercise:The calculation of minors and cofactors begins by focusing on each element one at a time. For instance, to compute the minor of the element \( a_{ij} \), you remove its row and column and calculate the determinant of the remaining submatrix.The clarity in recognizing which element is where aids in both manual and computational executions of matrix operations. It enhances your ability to find solutions effectively and comprehend more complex mathematics as these become the building blocks of larger systems.