Matrix Inverses: Unraveling S & T's Relationship

Hey math enthusiasts! Let's dive into the fascinating world of matrices and their inverses. Today, we're tackling a classic problem: determining if two given matrices are inverses of each other. This is a fundamental concept in linear algebra, and understanding it is key to unlocking more advanced topics. So, buckle up, grab your coffee (or your favorite beverage), and let's get started!

Understanding Matrix Inverses: The Basics

Alright, guys, before we jump into the matrices, let's refresh our memory on what matrix inverses are all about. Think of a matrix inverse as the mathematical equivalent of a reciprocal for numbers. When you multiply a number by its reciprocal, you get 1. Similarly, when you multiply a matrix by its inverse, you get the identity matrix. The identity matrix is a special square matrix that, when multiplied by any other matrix of the same size, leaves that matrix unchanged. It's like the number 1 in the world of matrices.

Now, how do you know if a matrix has an inverse? Not all matrices do! A matrix must be square (meaning it has the same number of rows and columns) and have a non-zero determinant to be invertible. The determinant is a scalar value that can be calculated from the elements of a square matrix. It provides valuable information about the matrix, including whether it has an inverse. If the determinant is zero, the matrix is singular and doesn't have an inverse. Finding the inverse of a 2x2 matrix is relatively straightforward. Let's say we have a matrix like this: A = [[a, b], [c, d]]. The inverse, denoted as A⁻¹, can be calculated using the following formula: A⁻¹ = (1 / (ad - bc)) * [[d, -b], [-c, a]]. The term (ad - bc) is the determinant of the matrix A. So, in order to calculate an inverse, you first need to confirm that the determinant of the matrix is non-zero, then compute the values and make sure they match with the options given in the problem statement.

So, why are matrix inverses so important? Well, they're essential for solving systems of linear equations. They allow us to isolate variables and find solutions. They also play a crucial role in various fields, including computer graphics, physics, and engineering. Think about it: a system of linear equations can represent anything from the forces acting on a bridge to the movement of objects in a video game. Having the ability to manipulate and solve these equations using matrices and their inverses is a powerful tool.

Analyzing Matrices S and T: Step-by-Step

Okay, guys, let's get down to the nitty-gritty and analyze our given matrices, S and T. We've got:

• S = \begin{bmatrix} 4 & 11
  -3 & -8 \end{bmatrix}

• T = \begin{bmatrix} -8 & -11
  3 & 4 \end{bmatrix}

Our goal is to determine if these two matrices are inverses of each other. The key here is to remember the definition of matrix inverses: If S and T are inverses, then their product (in either order, since matrix multiplication isn't always commutative) should equal the identity matrix, I. The identity matrix for a 2x2 matrix looks like this: I = \begin{bmatrix} 1 & 0
0 & 1 \end{bmatrix}. So, we need to calculate S * T and T * S to see if either product results in the identity matrix.

Let's start by calculating S * T:
S * T = \begin{bmatrix} 4 & 11
-3 & -8 \end{bmatrix} * \begin{bmatrix} -8 & -11
3 & 4 \end{bmatrix}

To multiply matrices, we take the dot product of the rows of the first matrix with the columns of the second matrix. The dot product is calculated by multiplying corresponding entries and summing the results. The element in the first row and first column of the resulting matrix is calculated as follows: (4 * -8) + (11 * 3) = -32 + 33 = 1. The element in the first row and second column is: (4 * -11) + (11 * 4) = -44 + 44 = 0. The element in the second row and first column: (-3 * -8) + (-8 * 3) = 24 - 24 = 0. And finally, the element in the second row and second column: (-3 * -11) + (-8 * 4) = 33 - 32 = 1. So, S * T = \begin{bmatrix} 1 & 0
0 & 1 \end{bmatrix}, which is indeed the identity matrix. Now, just to be sure, let's also calculate T * S.

T * S = \begin{bmatrix} -8 & -11
3 & 4 \end{bmatrix} * \begin{bmatrix} 4 & 11
-3 & -8 \end{bmatrix}

Doing the same dot product calculations, we get: (-8 * 4) + (-11 * -3) = -32 + 33 = 1; (-8 * 11) + (-11 * -8) = -88 + 88 = 0; (3 * 4) + (4 * -3) = 12 - 12 = 0; (3 * 11) + (4 * -8) = 33 - 32 = 1. Therefore, T * S = \begin{bmatrix} 1 & 0
0 & 1 \end{bmatrix}, which is also the identity matrix.

Conclusion: Which Statement is True?

Alright, after performing our matrix multiplications, we've found that both S * T and T * S equal the identity matrix. This means that matrices S and T are indeed inverses of each other. Thus, the statement that accurately describes the relationship between S and T is that they are inverses of each other. None of the other options would be correct, as the computations above demonstrate the inverse relationship. That means that the correct answer is the matrices S and T are inverses of each other. This is how we can determine if matrices are inverses of each other! So, you guys, remember this method, and you'll be able to solve similar problems with confidence. Keep practicing, and you'll become matrix masters in no time.

Now, you're all set to tackle similar problems. Just remember the key concepts: the definition of matrix inverses, the identity matrix, and how to perform matrix multiplication. Keep practicing, and you'll become matrix masters in no time! So, keep exploring the wonders of mathematics, and I'll catch you in the next one! Adios!