Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{lll} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array}\right] ; B=\left[\begin{array}{lll} 1 & -2 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

A and B are not inverses, as neither AB nor BA equals the identity matrix.

Step 1: Understand the Problem Statement

We need to determine if matrices \(A\) and \(B\) are inverses. Two matrices are inverses if their product, both \(AB\) and \(BA\), results in the identity matrix of the same dimension. Here, we will compute \(AB\) and \(BA\) and check if both equal the 3x3 identity matrix.

Step 2: Calculate AB

Multiply matrix \(A\) by matrix \(B\):\[A = \begin{bmatrix} 1 & 2 & 0 \ 0 & 1 & 0 \ 0 & 1 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -2 & 0 \ 0 & 1 & 0 \ 0 & -1 & 1 \end{bmatrix}\]The product \(AB\) is:\[AB = \begin{bmatrix}1(1) + 2(0) + 0(0) & 1(-2) + 2(1) + 0(-1) & 1(0) + 2(0) + 0(1) \0(1) + 1(0) + 0(0) & 0(-2) + 1(1) + 0(-1) & 0(0) + 1(0) + 0(1) \0(1) + 1(0) + 0(0) & 0(-2) + 1(1) + 0(-1) & 0(0) + 1(0) + 0(1) \end{bmatrix}\]Calculating each element:\[= \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 1 & 0 \end{bmatrix}\]

Step 3: Calculate BA

Multiply matrix \(B\) by matrix \(A\):\[B = \begin{bmatrix} 1 & -2 & 0 \ 0 & 1 & 0 \ 0 & -1 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 1 & 2 & 0 \ 0 & 1 & 0 \ 0 & 1 & 0 \end{bmatrix}\]The product \(BA\) is:\[BA = \begin{bmatrix}1(1) + -2(0) + 0(0) & 1(2) + -2(1) + 0(1) & 1(0) + -2(0) + 0(0) \0(1) + 1(0) + 0(0) & 0(2) + 1(1) + 0(1) & 0(0) + 1(0) + 0(0) \0(1) + -1(0) + 1(0) & 0(2) + -1(1) + 1(1) & 0(0) + -1(0) + 1(0) \end{bmatrix}\]Calculating each element:\[= \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{bmatrix}\]

Step 4: Conclusion

Check if both \(AB\) and \(BA\) are equal to the identity matrix. The identity matrix for 3x3 is \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\). \(AB\) is not equal to the identity matrix because its third row is not correct. \(BA\) is also not equal to the identity matrix because it has a zero in place of '1' in the (3,3) position. Hence, \(A\) and \(B\) are not inverses.

Key Concepts

Invertible Matrices

In the world of matrix algebra, an invertible matrix is a square matrix that possesses an inverse. Essentially, a matrix is deemed invertible if there exists another matrix which, when multiplied with the original, results in the identity matrix. Let's call matrix \(A\) and \(B\) two matrices. If \(AB = I\) and \(BA = I\), where \(I\) is the identity matrix, then \(A\) and \(B\) are inverses of each other. In simpler terms, multiplying them gives you a matrix that doesn’t change any other matrix when multiplied (like multiplying by 1 in regular numbers).

Invertible matrices are crucial in solving matrix equations. They allow you to "undo" matrix operations, providing solutions to systems of linear equations. However, not every matrix has an inverse, much like how dividing by zero is undefined in arithmetic. A matrix without an inverse is called singular or noninvertible. Understanding these concepts is key to navigating more complex topics in linear algebra and matrix theory.

Identity Matrix

The identity matrix is a fundamental component in matrix multiplication and matrix algebra. It is essentially the matrix equivalent of the number '1' in arithmetic. When you multiply any matrix by the identity matrix, it leaves the original matrix unchanged.

An identity matrix is a square matrix that has '1's on the diagonal from top-left to bottom-right and '0's in all other positions. For example, a 3x3 identity matrix looks like this:

• \[ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

When you perform matrix multiplication with an identity matrix, you are essentially taking the original matrix and producing a product that is exactly the same as the matrix you started with. This property makes the identity matrix uniquely important when determining if matrices are inverses of one another.

If two matrices \(A\) and \(B\) are such that their product, in both orders \(AB\) and \(BA\), results in the identity matrix, they are called inverses. This process is analogous to reversing a mathematical action, like multiplying by \(-1\) or \(\frac{1}{x}\) in arithmetic.

Matrix Algebra

Matrix algebra is a powerful mathematical framework that provides a way to manipulate and solve complex linear equations and data structures. It involves operations like addition, subtraction, and importantly, multiplication.

Understanding matrix multiplication is crucial because it lays the groundwork for more advanced concepts such as eigenvalues and transformations. Unlike arithmetic multiplication, matrix multiplication involves a row-by-column rule and requires that the number of columns in the first matrix equals the number of rows in the second matrix in order to produce a valid product.

In solving problems, such as determining if two matrices are inverses, matrix multiplication is the main operation used. The goal often is to see if combining the matrices produces the identity matrix.

• This involves careful computation of each element of the resulting matrix by summing the products of corresponding elements.

It's also essential for students to learn the difference between commutative and non-commutative properties within matrix operations, as generally \(AB eq BA\) in most cases. That's why checking both \(AB\) and \(BA\) is necessary for confirming matrix inverses. As students delve deeper, matrix algebra becomes the building block for various applied mathematics, including computer graphics, physics simulations, and more.