Problem 89 - A Graphical Approach to Precalculus with Limits

Question

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. $$(c A) d=(c d) A$$

Step-by-Step Solution

Answer

The equality \((cA)d = (cd)A\) holds true for any real scalars and matrices.

Step 1: Understanding the Exercise

The statement we need to show is \[(cA)d = (cd)A\]for any real number matrix \(A\) and scalars \(c\) and \(d\). This requires understanding of scalar multiplication in matrices.

Step 2: Matrix Scalar Multiplication

When multiplying a matrix by a scalar, each element of the matrix is multiplied by that scalar. Thus, for the matrix \(A\) and scalar \(c\), the product \(cA\) is:\[cA = \begin{bmatrix} ca_{11} & ca_{12} \ ca_{21} & ca_{22} \end{bmatrix}\]

Step 3: Scaling Twice: (cA)d

Now, let's consider multiplying matrix \(cA\) by scalar \(d\). Each element of the matrix \(cA\) gets multiplied by \(d\), which results in:\[(cA)d = \begin{bmatrix} (cd)a_{11} & (cd)a_{12} \ (cd)a_{21} & (cd)a_{22} \end{bmatrix} \]

Step 4: Direct Scaling: (cd)A

Similarly, consider multiplying the matrix \(A\) directly by the product of the scalars \(cd\). The product \((cd)A\) yields:\[(cd)A = \begin{bmatrix} (cd)a_{11} & (cd)a_{12} \ (cd)a_{21} & (cd)a_{22} \end{bmatrix}\]

Step 5: Equality Demonstration

It's clear that both resulting matrices \[(cA)d = \begin{bmatrix} (cd)a_{11} & (cd)a_{12} \ (cd)a_{21} & (cd)a_{22} \end{bmatrix}\] and \[(cd)A = \begin{bmatrix} (cd)a_{11} & (cd)a_{12} \ (cd)a_{21} & (cd)a_{22} \end{bmatrix}\]are identical, which shows that:\[(cA)d = (cd)A\]

Key Concepts

Matrix Multiplication

Matrix multiplication involves combining two matrices to form a new matrix. To perform this, you need to follow a rule that is unlike regular multiplication. Instead, each element of the resulting matrix is the sum of the products of corresponding elements of the rows of the first matrix and columns of the second matrix.
This process can only be done if the number of columns in the first matrix matches the number of rows in the second matrix. For example, if matrix A is of size \(2 \times 3\), it can only be multiplied by another matrix B that has \(3 \times n\) dimensions.

• One common point is: matrix multiplication is not commutative, meaning \(AB eq BA\) in general.
• Matrix multiplication is associative, which implies \((AB)C = A(BC)\).

Mastering matrix multiplication requires practice, especially in following the precise rules that govern how each row and column combination contributes to the final product.

Scalar Multiplication

Scalar multiplication in the context of matrices is straightforward. Unlike matrix multiplication, scalar multiplication involves multiplying every element of a matrix by a single number, known as a scalar.
For a matrix \(A\) of size \(2 \times 2\) and a scalar \(c\), the operation \(cA\) adjusts each element of the matrix \(A\) by the factor \(c\). This multiplication scales up or scales down the entire matrix, affecting its magnitude but not its direction.

• Scalar multiplication is distributive over matrix addition: \(c(A + B) = cA + cB\).
• It is also associative with other scalars: \((cd)A = c(dA)\).

This concept is vital when working with linear transformations and in many matrix algorithms where scaling is required.

Matrix Algebra

Matrix algebra comprises operations that involve matrices and the rules that govern these operations. It is akin to regular algebra, but specifically for matrices. In matrix algebra, you will encounter operations like addition, subtraction, multiplication, and the finding of inverses.
Matrix algebra is foundational in areas such as computer graphics, statistics, and even complex systems modeling.

• Addition and subtraction in matrix algebra are element-wise operations, meaning each element in the result is derived from adding or subtracting corresponding elements in the operand matrices.
• Inverse matrices, when multiplied by their original matrices, result in identity matrices, a key concept in solving linear systems of equations.

Understanding matrix algebra opens up a wide range of applications in both theoretical and practical areas of mathematics and beyond.