Problem 90 - 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. $$(c d) \mathcal{A}=c(d A)$$

Step-by-Step Solution

Answer

The statement is true as \((c d) A = c(d A)\) for \(2 \times 2\) matrices.

Step 1: Understand Matrix Multiplication by a Scalar

In matrix multiplication by a scalar, every element of the matrix is multiplied by the scalar. For a matrix \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \), multiplying by a scalar \( c \) gives \( cA = \begin{bmatrix} ca_{11} & ca_{12} \ ca_{21} & ca_{22} \end{bmatrix} \).

Step 2: Calculate \(dA\) for Matrix \(A\)

Multiply each element of the matrix \( A \) by the scalar \( d \), resulting in \( dA = \begin{bmatrix} d \, a_{11} & d \, a_{12} \ d \, a_{21} & d \, a_{22} \end{bmatrix} \).

Step 3: Calculate \(c(dA)\)

Now, multiply each element of the matrix \( dA \) by the scalar \( c \). This gives \( c(dA) = c \begin{bmatrix} d \, a_{11} & d \, a_{12} \ d \, a_{21} & d \, a_{22} \end{bmatrix} = \begin{bmatrix} c \, d \, a_{11} & c \, d \, a_{12} \ c \, d \, a_{21} & c \, d \, a_{22} \end{bmatrix} \).

Step 4: Calculate \((cd)A\)

Multiply the matrix \( A \) directly by the combined scalar \( cd \). Thus, \( (cd)A = \begin{bmatrix} (cd) \, a_{11} & (cd) \, a_{12} \ (cd) \, a_{21} & (cd) \, a_{22} \end{bmatrix} \).

Step 5: Compare \(c(dA)\) and \((cd)A\)

Both \( c(dA) \) and \( (cd)A \) result in the same matrix: \[ \begin{bmatrix} c \, d \, a_{11} & c \, d \, a_{12} \ c \, d \, a_{21} & c \, d \, a_{22} \end{bmatrix} \] This confirms that the operation yields the same outcome, proving that \((c d) A = c(d A)\).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in matrix algebra. It involves multiplying each element of a matrix by a single number, called a scalar. This operation is crucial for transforming matrices in various mathematical computations and can be seen in applications ranging from physics to economics.

To illustrate, consider a matrix \( A \) and a scalar \( c \). If \( A \) is \( 2 \times 2 \) matrix given by:

• \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \)

Multiplying it by a scalar \( c \) would yield:

• \( cA = \begin{bmatrix} ca_{11} & ca_{12} \ ca_{21} & ca_{22} \end{bmatrix} \)

Each element in matrix \( A \) is multiplied by \( c \), changing the scale of the matrix without altering its inherent structure. This is similar to how scaling works in geometry, where every point on the shape is moved proportionately from a point of origin.

Multiplying a matrix by a scalar is straightforward but fundamental, as it sets the stage for more complex operations like matrix addition, subtraction, and multiplication with other matrices.

2x2 Matrices

When we talk about matrices, a \(2 \times 2\) matrix is one of the simplest forms, having two rows and two columns. These matrices are not only simple but incredibly useful in solving linear equations, transformations, and even representing basic shifts in computer graphics.

A generic form of a \(2 \times 2\) matrix \( A \) can be represented as:

• \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \)

The compact size of a \(2 \times 2\) matrix makes it easy to visualize and perform operations on. When multiplying two \(2 \times 2\) matrices, follow the rule: the element at position \((i, j)\) in the result is obtained by the dot product of row \(i\) of the first matrix and column \(j\) of the second.

For many students, a \(2 \times 2\) matrix serves as a gateway to understanding larger matrices and complex matrix operations. With these matrices, properties and transformations become easier to comprehend.

Matrix Properties

Matrices satisfy several important properties that allow for their versatile application. Understanding these properties is key to manipulating matrices effectively in various mathematical and scientific contexts.

Here are some fundamental properties:

• Associative Property: The order in which you multiply matrices doesn’t change the result. For matrices \( A \), \( B \), and \( C \), this means \((AB)C = A(BC)\).
• Commutative Property of Scalar Multiplication: When a matrix is multiplied by a scalar, it behaves like regular arithmetic, so \( c(dA) = (cd)A \).
• Identity Matrix: Multiplying any matrix by an identity matrix \( I \) leaves it unchanged, where \( I \) for a \(2 \times 2\) matrix is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

The clarity of matrix properties simplifies operations, makes predicting results straightforward, and ensures consistency across applied mathematical problems.