Problem 87 - 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. \(c(A+B)=c A+c B,\) for any real number \(c\).

Step-by-Step Solution

Answer

The statement \(c(A+B)=cA+cB\) is true for any real number \(c\).

Step 1: Define Scalar Multiplication

Let's first understand what scalar multiplication is. If we multiply each element of matrix \(A\) by a scalar \(c\), the resulting matrix \(cA\) is given by: \[cA = \begin{bmatrix}ca_{11} & ca_{12} \ ca_{21} & ca_{22}\end{bmatrix}\] Similarly, for matrix \(B\), \[cB = \begin{bmatrix}cb_{11} & cb_{12} \ cb_{21} & cb_{22}\end{bmatrix}\]

Step 2: Matrix Addition

Next, we need to understand matrix addition. To add two matrices \(A\) and \(B\), we add the corresponding elements: \[A + B = \begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22}\end{bmatrix}\]

Step 3: Calculate \(c(A+B)\)

Now substitute the result of \(A + B\) into the scalar multiplication formula:\[c(A + B) = c \begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22}\end{bmatrix}\]Apply the scalar multiplication:\[c(A + B) = \begin{bmatrix}c(a_{11} + b_{11}) & c(a_{12} + b_{12}) \ c(a_{21} + b_{21}) & c(a_{22} + b_{22})\end{bmatrix}\]

Step 4: Expand \(cA + cB\)

Now, expand \(cA + cB\):\[cA = \begin{bmatrix}ca_{11} & ca_{12} \ ca_{21} & ca_{22}\end{bmatrix}, \quad cB = \begin{bmatrix}cb_{11} & cb_{12} \ cb_{21} & cb_{22}\end{bmatrix}\]Add them together:\[cA + cB = \begin{bmatrix}ca_{11} + cb_{11} & ca_{12} + cb_{12} \ ca_{21} + cb_{21} & ca_{22} + cb_{22}\end{bmatrix}\]

Step 5: Compare Results

Compare the matrices from Step 3 and Step 4:\[c(A + B) = \begin{bmatrix}c(a_{11} + b_{11}) & c(a_{12} + b_{12}) \ c(a_{21} + b_{21}) & c(a_{22} + b_{22})\end{bmatrix} \]\[cA + cB = \begin{bmatrix}ca_{11} + cb_{11} & ca_{12} + cb_{12} \ ca_{21} + cb_{21} & ca_{22} + cb_{22}\end{bmatrix}\]Since \(c(a_{ij} + b_{ij}) = ca_{ij} + cb_{ij}\), both matrices are equal.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra which involves adding corresponding elements of two matrices. To perform matrix addition, both matrices need to have the same dimensions, which allows each element in one matrix to align with a corresponding element in the second matrix. For example, given two matrices:

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

To compute the sum A + B, add each element in matrix A to the respective element in matrix B:\[A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}\]Remember, matrix addition is straightforward if you align the matrices well, keeping in mind their dimensions have to be identical.

Property of Matrices

The properties of matrices often simplify complex operations and prove various matrix equations like the distributive property, which states that scalar multiplication distributes over matrix addition. This can be expressed as:
\[c(A + B) = cA + cB\]
What this implies is that if you first add two matrices, A and B, and then multiply the result by a scalar c, it will yield the same result as if you first multiplied each individual matrix by the scalar and then added those products. This property is crucial in simplifying computations and verifying the correctness of matrix operations.
For example, applying the scalar multiplication to each element after adding matrices A and B, or individually to A and B before adding, the resulting matrices are identical. The distributive property underlines the flexibility and predictability of mathematical operations involving matrices.

2x2 Matrices

2x2 matrices are one of the simplest forms of matrices used in mathematics, making them an ideal choice for learning and teaching basic matrix operations such as addition, subtraction, and multiplication by a scalar. Each 2x2 matrix is structured as:
\[\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\]
Despite their simplicity, they are immensely useful and appear in diverse applications ranging from solving systems of equations to transformations in computer graphics. Their manageability makes them perfect for beginners to practice essential matrix algebra skills.
In mathematical exercises, 2x2 matrices are often used to demonstrate fundamental properties such as commutativity of addition or distributive properties of scalars, serving as a building block for understanding larger and more complex matrices. The ability to quickly perform operations on 2x2 matrices sets a robust foundation for progressing towards complex matrix algebraic concepts.