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

Step-by-Step Solution

Answer

The property \(c(A+B) = cA + cB\) holds for 2x2 matrices.

Step 1: Matrix Addition

First, compute the sum of matrices \(A\) and \(B\). The elements of the resulting matrix \(A + B\) are obtained by adding the corresponding elements of \(A\) and \(B\). Thus, \(A+B = \left[\begin{array}{cc}a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22}\end{array}\right]\).

Step 2: Multiply by Scalar

Multiply the matrix \(A+B\) by the scalar \(c\). Each element of the matrix \(A+B\) is multiplied by \(c\). Therefore, \(c(A+B) = \left[\begin{array}{cc}c(a_{11} + b_{11}) & c(a_{12} + b_{12}) \ c(a_{21} + b_{21}) & c(a_{22} + b_{22})\end{array}\right]\).

Step 3: Scalar Distribution

Now, multiply each matrix \(A\) and \(B\) individually by \(c\). This gives \(cA = \left[\begin{array}{cc}c a_{11} & c a_{12} \ c a_{21} & c a_{22}\end{array}\right]\) and \(cB = \left[\begin{array}{cc}c b_{11} & c b_{12} \ c b_{21} & c b_{22}\end{array}\right]\).

Step 4: Add Scaled Matrices

Add the results obtained from multiplying \(A\) and \(B\) by the scalar \(c\). We compute \(cA + cB = \left[\begin{array}{cc}c a_{11} + c b_{11} & c a_{12} + c b_{12} \ c a_{21} + c b_{21} & c a_{22} + c b_{22}\end{array}\right]\).

Step 5: Verify Equality

Match the expressions from Steps 2 and 4 to verify they are identical: \( \left[\begin{array}{cc}c(a_{11} + b_{11}) & c(a_{12} + b_{12}) \ c(a_{21} + b_{21}) & c(a_{22} + b_{22})\end{array}\right] = \left[\begin{array}{cc}c a_{11} + c b_{11} & c a_{12} + c b_{12} \ c a_{21} + c b_{21} & c a_{22} + c b_{22}\end{array}\right]\). Since the two expressions are the same, it confirms the property: \( c(A+B) = cA + cB \).

Key Concepts

Scalar Multiplication

Scalar multiplication in matrix algebra involves multiplying each element of a matrix by a scalar, which is a real number. Imagine you have a matrix that looks like this: \[ M = \begin{bmatrix} m_{11} & m_{12} \ m_{21} & m_{22} \end{bmatrix} \] With a scalar value of \( c \), scalar multiplication will transform the matrix into a new one where each element is multiplied by \( c \):- The element at position (1,1) becomes \( c \times m_{11} \).- The element at position (1,2) becomes \( c \times m_{12} \).- The element at position (2,1) becomes \( c \times m_{21} \).- The element at position (2,2) becomes \( c \times m_{22} \).This process is straightforward but crucial when working with matrices. It maintains the same dimensions of the original matrix. Scalar multiplication is often utilized in various matrix operations to scale the values of the entire matrix without changing its structure.
A helpful way to think about scalar multiplication is like adjusting the brightness of an image, where every pixel becomes brighter or darker based on a uniform factor.

Matrix Addition

Matrix addition is the process of adding two matrices of the same dimensions by adding the corresponding elements. Given matrices \[ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \] and \[ B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \], their sum \( A + B \) results in a new matrix - The element at the first row, first column becomes \( a_{11} + b_{11} \).- The element at the first row, second column becomes \( a_{12} + b_{12} \).- The element at the second row, first column becomes \( a_{21} + b_{21} \).- The element at the second row, second column becomes \( a_{22} + b_{22} \).Matrix addition is simple but only possible when the matrices involved have the exact same dimensions. This operation is similar to adding numbers; however, it's done element by element. Matrix addition lays the foundational brick for more complex operations in matrix algebra, setting the stage for understanding linear transformations and systems of equations.

Distributive Property

The distributive property in matrix algebra is a crucial rule that relates scalar multiplication and matrix addition. It states that a scalar multiplied by the sum of two matrices is equal to the sum of the scalar multiplied by each of the matrices individually:\[ c(A + B) = cA + cB \] where \( c \) is a scalar, and \( A \) and \( B \) are matrices.To break it down:- Compute the sum \( A + B \), resulting in a matrix where each element is the sum of the corresponding elements of \( A \) and \( B \).- Perform scalar multiplication \( c \times (A + B) \) on the resulting matrix.- Alternatively, multiply each matrix by \( c \) separately to get \( cA \) and \( cB \), and then add them together.This property demonstrates flexibility in computation and highlights the powerful properties of matrix operations, paralleling the distributive property of numbers. It aids in simplifying expressions and proving various matrix algebra identities, vital for larger computations and theoretical proofs.

Real Numbers

Real numbers are the building blocks for many mathematical concepts including matrix algebra. These numbers include all the possible types of numbers you encounter in everyday calculations: - **Whole numbers** like 1, 2, and 3. - **Fractions** such as 1/2 or 3/4. - **Decimals** like 0.75 and 2.5. - **Negative numbers** such as -1, -2.3, etc. In matrix algebra, every element within a matrix is a real number, ensuring a wide range of flexibility and application in various mathematical models and real-life problems. The set of real numbers is immense, stretching from negative infinity to positive infinity. Still, they're governed by specific arithmetic rules crucial when performing matrix operations like addition, multiplication, and maintaining consistent dimensions throughout matrix transformations.