Problem 28 - Algebra for College Students

Question

For Problems \(27-30\), use the following matrices. \(\begin{aligned} &A=\left[\begin{array}{rr} -2 & 3 \\ 5 & 4 \end{array}\right] \quad B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right] \quad D=\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right]\\\ &I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}\) $$ \text { Compute } A C \text { and } C A \text {. } $$

Step-by-Step Solution

Answer

The matrix products are \(AC = \begin{bmatrix} 1 & 0 \\ 9 & 0 \end{bmatrix}\) and \(CA = \begin{bmatrix} -2 & 3 \\ -2 & 3 \end{bmatrix}\).

Step 1: Matrix Multiplication Basics

To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Here, both matrix \(A\) and \(C\) are 2x2 matrices, so both products \(AC\) and \(CA\) are defined.

Step 2: Compute Matrix Product AC

To find \(AC\), multiply each element of the rows of \(A\) with the corresponding elements of the columns of \(C\) and sum the products. The calculations for matrix \(AC\) will be:\[AC = \begin{bmatrix}(-2) \cdot 1 + 3 \cdot 1 & (-2) \cdot 0 + 3 \cdot 0 \5 \cdot 1 + 4 \cdot 1 & 5 \cdot 0 + 4 \cdot 0\end{bmatrix}\]This simplifies to:\[AC = \begin{bmatrix}1 & 0 \9 & 0\end{bmatrix}\]

Step 3: Compute Matrix Product CA

To find \(CA\), multiply each element of the rows of \(C\) with the corresponding elements of the columns of \(A\) and sum the products. The calculations for matrix \(CA\) will be:\[CA = \begin{bmatrix}1 \cdot (-2) + 0 \cdot 5 & 1 \cdot 3 + 0 \cdot 4 \1 \cdot (-2) + 0 \cdot 5 & 1 \cdot 3 + 0 \cdot 4\end{bmatrix}\]This simplifies to:\[CA = \begin{bmatrix}-2 & 3 \-2 & 3\end{bmatrix}\]

Key Concepts

2x2 matrices

Matrices are a core concept in algebra that allow us to organize and manipulate numbers. A 2x2 matrix is a simple square matrix with two rows and two columns. This structure, when expressed in a matrix format, looks like:
\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] In this format, the elements of the matrix are denoted by \(a\), \(b\), \(c\), and \(d\). Each position in the matrix carries unique information, and these can be coefficients or constants based on the problem you are solving.
2x2 matrices are used in various mathematical computations in fields such as geometry, physics, and computer science. Learning to manipulate these matrices is essential for performing more complex matrix computations.

matrix product

The matrix product is an extensive operation performed on two matrices where, typically, the elements are multiplied and summed according to specific rules. In our exercise, we are working with 2x2 matrices, where the matrix product involves combining two matrices, say, \( A \) and \( C \), to produce a new matrix, referred to as \( AC \).
To compute the matrix product, you take each row of the first matrix and multiply it with each column of the second matrix, adding up the products to get a single number for the resultant matrix. This operation must follow:

• The number of columns in the first matrix must match the number of rows in the second matrix.
• The result is another matrix, often with a size defined by the rows of the first matrix and columns of the second matrix.

For our 2x2 matrices \( AC \) may look structurally identical to \( A \) and \( C \) but contains entirely different numerical values derived from these computations. Understanding the logic behind each calculation step is crucial in matrix product operations.

matrix computation

Matrix computations are a series of mathematical operations you can perform on matrices such as addition, multiplication, and finding the determinant. These operations allow you to solve complex problems in both pure and applied mathematics. When focusing on matrix multiplication, which we did for matrices \( A \) and \( C \), you derive new matrices and gain insights into systems they represent.
Consider the computational steps to determine \( AC \):

• The first row multiplied by the first column gives the top left element.
• The first row multiplied by the second column gives the top right element.
• And similarly, for the second row multiplied by the first and second columns for the bottom elements.

Every calculation introduces you to how matrices interact and compute larger expressions. Mastery of these computations equips you for higher algebraic concepts that rely on matrix operations.

algebra

Algebra is more than solving for a single variable; it encompasses expressions like matrices. In this realm, matrices symbolize sets of equations, transformations, or networks and enhance algebraic problem-solving through compact, structured methods.
With matrices, rather than handling one equation at a time, you work with groups simultaneously, leading to efficient processing and insight discovery in multi-variable situations. Whether you're multiplying 2x2 matrices or exploring other algebraic forms, the principles of organization and manipulation remain similar, making algebra a flexible, potent tool in mathematics.
Grasping algebra through matrices introduces a systematic way to handle equations and builds a fundamental understanding beneficial across various scientific and mathematical fields. As complexity increases, algebraic skills developed in matrix operations remain relevant, aiding in mathematical analysis and solutions.