Question

Find all (a) minors and (b) cofactors of the matrix. $$\left[\begin{array}{rr}3& 1\\ -2& -4\end{array}\right]$$

Short answer

The minors of the matrix are $$\left[\begin{array}{cc}-4& -2\\ 1& 3\end{array}\right]$$, while the cofactors of the matrix are $$\left[\begin{array}{cc}-4& 2\\ -1& 3\end{array}\right]$$.

Step-by-step solution

Minor Calculation

Firstly, calculate minors for each entries in the given 2x2 matrix. The minor $m_{11}$ of 3 is the determinant obtained by removing the first row and first column, which gives the element -4. Similarly, the minor $m_{12}$ of 1 is the determinant gained by removing the first row and second column, which gives -2. The minor $m_{21}$ of -2 is the determinant after removing the second row and first column, which gives 1. Lastly, the minor $m_{22}$ of -4 is the determinant obtained by removing the second row and second column, which gives 3.

Cofactor Calculation

Then, we calculate the cofactors for each entries in the matrix. The cofactor $c_{11}$ of element 3 (at position 1,1) is $m_{11}\times (-1{)}^{1+1}=-4\times 1=-4$. The cofactor $c_{12}$ of 1 (at position 1,2) is $m_{12}\times (-1{)}^{1+2}=-2\times -1=2$. The cofactor $c_{21}$ of -2 (at position 2,1) is $m_{21}\times (-1{)}^{2+1}=1\times -1=-1$. The cofactor $c_{22}$ of -4 (at position 2,2) is $m_{22}\times (-1{)}^{2+2}=3\times 1=3$.

Results presentation

Present the results in a clear and understandable manner. The minors of the matrix are $$\left[\begin{array}{cc}-4& -2\\ 1& 3\end{array}\right]$$. On the other hand, the cofactors of the matrix are $$\left[\begin{array}{cc}-4& 2\\ -1& 3\end{array}\right]$$.

Key concepts

Determinant Calculation

The determinant of a 2x2 matrix is a numeric value that gives us insight into matrix properties such as singularity and area transformation. It is calculated by taking a matrix:
$$\begin{array}{ccc}a& bc& d\end{array}$$
and applying the formula $ad-bc$, where $a,b,c,d$ are elements of the matrix. In our exercise,
$$\begin{array}{rrr}3& 1\text{-}2& -4\end{array}$$
the determinant would be $(3)(-4)-(1)(-2)=-12+2=-10$.
For minors, which are determinants of submatrices, you simply exclude the row and column of the element of interest. Calculating minors is a fundamental step before finding cofactors and ultimately determining a matrix inverse, should it exist. Understanding how to compute minors and cofactors for a 2x2 matrix is essential when expanding these concepts to larger matrices.

Linear Algebra

Linear Algebra is a branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It's fundamental to various areas such as physics, engineering, computer science, and more. In the context of our exercise, Linear Algebra provides principles for understanding and operating with matrices.
The minor and cofactor calculations are crucial when we start dealing with matrix operations, such as finding the inverse of a matrix or solving systems of linear equations. These concepts are building blocks for more complex operations and applications in Linear Algebra. Developing a solid understanding of these basic operations with 2x2 matrices prepares you to handle higher-dimensional matrices with more confidence.

2x2 Matrix Operations

Operations with 2x2 matrices are the simplest to visualize and understand, making them great for introductory purposes. Besides determinant calculation, other operations include matrix addition, matrix multiplication, and finding the inverse, provided the determinant is not zero.
Minors play a role when one wants to find the inverse of a 2x2 matrix, where you first need to calculate the minors, then the cofactors, and finally transpose the matrix of cofactors and divide by the determinant.
The ability to perform these basic operations is a vital skill in Linear Algebra, as matrices are used to represent linear transformations. The concepts exercised in minors and cofactors of a 2x2 matrix extend to larger matrices, although the complexity increases. Thus, mastering such foundational operations empowers you to advance in understanding and contributing to various applications that utilize Linear Algebra.