Problem 88 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

Scalar multiplication is distributive over matrix addition.

Step 1: Compute (c + d)A

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

Step 2: Compute cA + dA

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

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

Step 3: Compare the two results

Each entry of \((c+d)A\) is \((c+d)a_{ij}\), and each entry of \(cA + dA\) is \(ca_{ij} + da_{ij}\).

By the distributive property of real numbers: \((c+d)a_{ij} = ca_{ij} + da_{ij}\) for all \(i, j\).

Therefore \((c+d)A = cA + dA\), proving that scalar multiplication distributes over scalar addition for \(2 \times 2\) matrices. \(\square\)

Key Concepts

Scalar Multiplication

Scalar multiplication in matrix algebra involves multiplying each element of a matrix by a single number, known as a scalar. This operation scales the entire matrix by that scalar value.

For example, consider a matrix \( A \) with elements \( a_{11}, a_{12}, a_{21}, \) and \( a_{22} \). When we multiply this matrix by a scalar \( c \), each element in the matrix is multiplied by \( c \). The resulting matrix \( cA \) becomes:

• \( cA = \left[ \begin{array}{ll} c \cdot a_{11} & c \cdot a_{12} \ c \cdot a_{21} & c \cdot a_{22} \end{array} \right] \)

Similarly, multiplying by another scalar \( d \), you obtain:

• \( dA = \left[ \begin{array}{ll} d \cdot a_{11} & d \cdot a_{12} \ d \cdot a_{21} & d \cdot a_{22} \end{array} \right] \)

This principle of scalar multiplication is foundational to understanding how matrices interact with real numbers in algebra.

2x2 Matrices

A 2x2 matrix is a simple yet powerful structure in matrix algebra. It consists of two rows and two columns, forming a square shape. A generic 2x2 matrix \( A \) is often written as:

• \( \left[ \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \end{array} \right] \)

Each element \( a_{ij} \) represents a real number, and the positions are indexed by their row \( i \) and column \( j \). Understanding how these matrices operate is crucial for further learning that extends to other mathematical concepts like systems of equations or transformations.

2x2 matrices are especially common in introductory linear algebra courses due to their simplicity and ease of computation. They serve as building blocks for larger and more complex matrices.

Distributive Property

The distributive property is a key rule in algebra that also applies to matrices. This property states that when you have two scalars, say \( c \) and \( d \), the expression \((c + d)A\) is the same as \( cA + dA \). Here's how it works with a 2x2 matrix \( A \):

Start by calculating \( (c + d)A \):

• \( (c+d)A = \left[ \begin{array}{cc} (c+d) \cdot a_{11} & (c+d) \cdot a_{12} \ (c+d) \cdot a_{21} & (c+d) \cdot a_{22} \end{array} \right] \)

Next, distribute \( A \) across \( c \) and \( d \):

• \( cA = \left[ \begin{array}{cc} c \cdot a_{11} & c \cdot a_{12} \ c \cdot a_{21} & c \cdot a_{22} \end{array} \right] \)
• \( dA = \left[ \begin{array}{cc} d \cdot a_{11} & d \cdot a_{12} \ d \cdot a_{21} & d \cdot a_{22} \end{array} \right] \)

Then, add \( cA \) and \( dA \) element-wise:

• \( cA + dA = \left[ \begin{array}{cc} c \cdot a_{11} + d \cdot a_{11} & c \cdot a_{12} + d \cdot a_{12} \ c \cdot a_{21} + d \cdot a_{21} & c \cdot a_{22} + d \cdot a_{22} \end{array} \right] \)

This shows mathematically how the distributive property helps in simplifying and solving matrix algebra problems. It's a fascinating aspect that reflects the harmony between numbers and structures.