Inverse Of Square Matrix M: A Step-by-Step Guide

Understanding Square Matrices and Their Inverses

In the fascinating world of mathematics, particularly within linear algebra, square matrices hold a special place. A square matrix is simply a matrix that has the same number of rows as it has columns. Think of it like a perfectly balanced scale; everything is symmetrical. These matrices are fundamental to solving systems of linear equations, representing transformations in geometry, and much more. But what makes a square matrix truly powerful is its potential to have an inverse. An inverse of a square matrix, often denoted as $M^{-1}$, is another matrix that, when multiplied by the original matrix $M$, yields the identity matrix ($I$). The identity matrix is like the number '1' in regular multiplication – it doesn't change anything when you multiply by it. For a 2x2 matrix, the identity matrix looks like $\left[\begin{array}{rr} 1 & 0 \ 0 & 1 \end{array}\right]$. Finding this inverse is crucial for tasks like solving for unknown variables in matrix equations, where dividing by a matrix isn't defined, but multiplying by its inverse is. This process of finding the inverse allows us to isolate variables and determine unique solutions. It's a key concept that unlocks deeper understanding and practical applications in various scientific and engineering fields. The existence of an inverse is not guaranteed for all square matrices; a matrix must be non-singular (meaning its determinant is not zero) to have an inverse. This article will guide you through the process of finding the inverse of a specific 2x2 square matrix, $M=\left[\begin{array}{rr} 1 & -3 \ 4 & 2 \end{array}\right]$, demonstrating the practical steps involved and the underlying mathematical principles.

Calculating the Determinant: The First Crucial Step

Before we can even think about finding the inverse of a square matrix, we absolutely must determine if an inverse even exists. The key to this lies in calculating the determinant of the matrix. For a 2x2 matrix like our example, $M=\left[\begin{array}{rr} a & b \ c & d \end{array}\right]$, the determinant, denoted as $\det(M)$ or $|M|$, is calculated using a simple formula: $ad - bc$. This value is more than just a number; it's a crucial indicator of the matrix's properties. If the determinant is zero, the matrix is called singular, and it means it does not have an inverse. This is because attempting to find an inverse for a singular matrix would involve division by zero at some point in the calculation, which is mathematically undefined. Conversely, if the determinant is non-zero, the matrix is non-singular, and we can proceed with confidence to find its unique inverse. So, let's roll up our sleeves and calculate the determinant for our specific matrix $M=\left[\begin{array}{rr} 1 & -3 \ 4 & 2 \end{array}\right]$. Here, $a=1$, $b=-3$, $c=4$, and $d=2$. Applying the formula, we get: $\det(M) = (1 \times 2) - (-3 \times 4)$. This simplifies to $2 - (-12)$, which equals $2 + 12 = 14$. Since our determinant is $14$, which is definitely not zero, we know that our matrix $M$ is non-singular and possesses a unique inverse. This calculation, though seemingly simple, is the gatekeeper to finding the inverse, ensuring that our subsequent steps will lead to a valid and meaningful result.

Constructing the Adjugate Matrix: A Step Towards the Inverse

Now that we've confirmed our matrix $M$ has an inverse by calculating a non-zero determinant, the next step in finding the inverse is to construct what's called the adjugate matrix (sometimes also referred to as the adjoint matrix). For a 2x2 matrix $M=\left[\begin{array}{rr} a & b \ c & d \end{array}\right]$, the adjugate matrix is found through a specific rearrangement of its elements. You swap the positions of the elements on the main diagonal (the 'a' and 'd' elements) and then negate the elements on the off-diagonal (the 'b' and 'c' elements). So, the adjugate of $M$, denoted as $\text{adj}(M)$, is given by the formula: $\text{adj}(M) = \left[\begin{array}{rr} d & -b \ -c & a \end{array}\right]$. Let's apply this to our matrix $M=\left[\begin{array}{rr} 1 & -3 \ 4 & 2 \end{array}\right]$. Here, $a=1$, $b=-3$, $c=4$, and $d=2$. Following the rule, we swap $a$ and $d$, so $2$ goes to the top-left and $1$ goes to the bottom-right. Then, we negate $b$ and $c$. Negating $-3$ gives us $3$, and negating $4$ gives us $-4$. Therefore, the adjugate matrix of $M$ is: $\text{adj}(M) = \left[\begin{array}{rr} 2 & 3 \ -4 & 1 \end{array}\right]$. This adjugate matrix is a vital component in the final calculation of the inverse. It's essentially a