Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Show that \(B\) is the inverse of \(A\). \(\begin{aligned} A &=\left[\begin{array}{rrrr}2 & 0 & 2 & 1 \\ 3 & 0 & 0 & 1 \\\ -1 & 1 & -2 & 1 \\ 3 & -1 & 1 & 0\end{array}\right] \\ B &=\frac{1}{3}\left[\begin{array}{rrrr}-1 & 3 & -2 & -2 \\ -2 & 9 & -7 & -10 \\\ 1 & 0 & -1 & -1 \\ 3 & -6 & 6 & 6\end{array}\right] \end{aligned}\)

Step-by-Step Solution

Answer

Matrix \(B\) is the inverse of matrix \(A\) if both products \(AB\) and \(BA\) result in the identity matrix. The identity matrix is a square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros.

Step 1: Calculate AB

First, multiply matrix \(A\) by matrix \(B\). The result is a new matrix which we'll call \(AB\). The element in row \(i\) and column \(j\) of \(AB\) is computed as the sum of the product of corresponding elements from row \(i\) of matrix \(A\) and column \(j\) of matrix \(B\).

Step 2: Check if AB is the identity matrix

After calculating \(AB\), check if the resulting matrix is the identity matrix. The identity matrix \(I\) is a square matrix with ones on the main diagonal and zeros elsewhere. So, the matrix \(AB\) equals \(\begin{bmatrix}1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1\end{bmatrix}\) if \(B\) is indeed the inverse of \(A\).

Step 3: Calculate BA

Next, multiply matrix \(B\) by matrix \(A\). The computation is done in the same way as in Step 1, but now with interchange of roles for \(A\) and \(B\). The resulting matrix is labeled as \(BA\).

Step 4: Check if BA is the identity matrix

If matrix \(BA\) equals the identity matrix, \(B\) is the inverse of \(A\). The identity matrix that we are comparing against is the same as in Step 2.

Key Concepts

Identity Matrix

An identity matrix is a special type of square matrix that plays a pivotal role in matrix theory, particularly when working with matrix inverses. Imagine it as the number one in matrix algebra; it is the matrix equivalent of multiplying by 1 in arithmetic.

Here’s what makes the identity matrix special:

• The main diagonal—the line of elements from the top left to the bottom right—consists of ones.
• All other positions in the matrix have zeros.

For example, a 4x4 identity matrix looks like this:

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

Multiplying any matrix by an identity matrix does not change the original matrix, similar to how any number multiplied by 1 is unchanged. Therefore, if the result of multiplying matrices A and B results in an identity matrix, we say B is the inverse of A.

Matrix Multiplication

Matrix multiplication is a key operation in matrix algebra that requires precision in calculations. Unlike regular multiplication, you cannot multiply just any two matrices; their dimensions must align.

Here are the rules:

• If matrix A is of size m x n and matrix B is of size n x p, we can multiply them.
• The resulting matrix, AB, will be of size m x p.

The process involves taking the corresponding elements from the rows of the first matrix and columns of the second matrix, multiplying them, and summing the results:

• For each element at position (i, j) in the resulting matrix, calculate the sum of the products of elements from row i of the first matrix and column j of the second matrix.

This process develops a matrix through systematic calculations, which need to be exact to ensure the desired outcome, such as verifying if B is indeed the inverse of A by resulting in the identity matrix.

Matrix Algebra

Matrix algebra encompasses several operations that allow us to manipulate and solve equations involving matrices. This branch of algebra provides the foundations for much of linear algebra and is integral in many fields such as physics, computer science, and engineering.

Key operations in matrix algebra include:

• Addition and Subtraction: Matrices can be added or subtracted element by element, provided they are of the same dimensions.
• Multiplication: As covered, requires specific dimensions to multiply corresponding elements.
• Inverses: If a matrix A has an inverse B, then A multiplied by B yields the identity matrix.
• Determinants and Eigenvalues: Determinants help in determining matrix invertibility, while eigenvalues assist in understanding matrix properties.

Using these operations, matrix algebra can solve complex systems of equations with clarity and precision. Thanks to these foundational ideas, matrices become powerful tools in computational mathematics.