Problem 88 - 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. \((c+d) A=c A+d A,\) for any real numbers \(c\) and \(d\).

Step-by-Step Solution

Answer

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

Step 1: Understand Matrix Multiplication by a Scalar

Matrix multiplication by a scalar means multiplying every element of the matrix by that scalar. For example, if \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \) and \( c \) is a scalar, then \( cA = \begin{bmatrix} c \cdot a_{11} & c \cdot a_{12} \ c \cdot a_{21} & c \cdot a_{22} \end{bmatrix} \). The same concept applies to any other scalar.

Step 2: Apply Scalar Multiplication to Each Matrix

Compute \( (c+d)A \), \( cA \), and \( dA \) separately. For \( (c+d)A \), multiply each element of \( A \) by \( c+d \):\((c+d)A = \begin{bmatrix} (c+d) \cdot a_{11} & (c+d) \cdot a_{12} \ (c+d) \cdot a_{21} & (c+d) \cdot a_{22} \end{bmatrix}\).For \( cA \), multiply each element of \( A \) by \( c \):\(cA = \begin{bmatrix} c \cdot a_{11} & c \cdot a_{12} \ c \cdot a_{21} & c \cdot a_{22} \end{bmatrix}\).For \( dA \), multiply each element of \( A \) by \( d \):\(dA = \begin{bmatrix} d \cdot a_{11} & d \cdot a_{12} \ d \cdot a_{21} & d \cdot a_{22} \end{bmatrix}\).

Step 3: Add the Results of cA and dA

Add the matrices \( cA \) and \( dA \) element by element:\(cA + dA = \begin{bmatrix} (c \cdot a_{11} + d \cdot a_{11}) & (c \cdot a_{12} + d \cdot a_{12}) \ (c \cdot a_{21} + d \cdot a_{21}) & (c \cdot a_{22} + d \cdot a_{22}) \end{bmatrix}\).Simplify this by factoring out common factors:\(cA + dA = \begin{bmatrix} (c+d) \cdot a_{11} & (c+d) \cdot a_{12} \ (c+d) \cdot a_{21} & (c+d) \cdot a_{22} \end{bmatrix}\).

Step 4: Compare the Results

We have:\((c+d)A = \begin{bmatrix} (c+d) \cdot a_{11} & (c+d) \cdot a_{12} \ (c+d) \cdot a_{21} & (c+d) \cdot a_{22} \end{bmatrix}\)and\(cA + dA = \begin{bmatrix} (c+d) \cdot a_{11} & (c+d) \cdot a_{12} \ (c+d) \cdot a_{21} & (c+d) \cdot a_{22} \end{bmatrix}\).Since both the expressions are the same, it is demonstrated that \((c+d)A = cA + dA\).

Key Concepts

Scalar Multiplication

Scalar multiplication is a straightforward yet important matrix operation. It involves multiplying every element of a matrix by a single number, known as the scalar. Say you have a matrix \(A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\) and a scalar \(c\). When you perform scalar multiplication by \(c\), every element in matrix \(A\) is multiplied by \(c\).
Thus, the matrix \(cA\) will look like this:

• First row becomes \(\begin{bmatrix} c \cdot a_{11} & c \cdot a_{12} \end{bmatrix}\)
• Second row becomes \(\begin{bmatrix} c \cdot a_{21} & c \cdot a_{22} \end{bmatrix}\)

This operation effectively "scales" every element in the matrix by the same factor. It's analogous to multiplying a vector or a set of numbers by a constant, stretching or shrinking it equivalently in all dimensions.

Matrix Addition

Matrix addition is another fundamental operation. It requires that the matrices you are adding have the same dimensions. Let's consider two matrices: \(A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\) and \(B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\). To add these matrices, you simply add corresponding elements from each matrix.
This means you will sum up elements from the first row and first column, the first row and second column, and so on. Here is the operation explained:

• The resulting matrix from adding \(A\) and \(B\) will be \(\begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}\)

Matrix addition is particularly helpful when dealing with transformations and combined effects in mathematics, physics, and engineering.

Distributive Property

The distributive property is a key concept when performing operations with matrices, just like in arithmetic with numbers. It tells us how scalar multiplication interacts with matrix addition. For matrices, this property can be seen as: \((c + d)A = cA + dA\) for any matrix \(A\) and scalars \(c\) and \(d\).
The beauty of this property lies in its simplicity and effectiveness. Consider:\[ (c+d)A = \begin{bmatrix} (c+d) \cdot a_{11} & (c+d) \cdot a_{12} \ (c+d) \cdot a_{21} & (c+d) \cdot a_{22} \end{bmatrix} \]
Using the distributive property, you can break this down into:

• First compute \(cA = \begin{bmatrix} c \cdot a_{11} & c \cdot a_{12} \ c \cdot a_{21} & c \cdot a_{22} \end{bmatrix}\)
• Then compute \(dA = \begin{bmatrix} d \cdot a_{11} & d \cdot a_{12} \ d \cdot a_{21} & d \cdot a_{22} \end{bmatrix}\)
• Add these: \(cA + dA = \begin{bmatrix} (c \cdot a_{11} + d \cdot a_{11}) & (c \cdot a_{12} + d \cdot a_{12}) \ (c \cdot a_{21} + d \cdot a_{21}) & (c \cdot a_{22} + d \cdot a_{22}) \end{bmatrix}\), simplifying to \((c+d)A\).

This property is essential for simplifying expressions and understanding complex linear transformations.