Problem 32 - Precalculus: Functions and Graphs

Question

Verify the identity for $$\boldsymbol{A}=\left[\begin{array}{ll}a & b \\\c & d\end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll}p & \boldsymbol{q} \\\r & s\end{array}\right], \quad \boldsymbol{C}=\left[\begin{array}{ll}\boldsymbol{w} & \boldsymbol{x} \\\\\boldsymbol{y} & z\end{array}\right]$$ and real numbers \(m\) and \(n\). $$(m+n) A=m A+n A$$

Step-by-Step Solution

Answer

The identity \((m+n)A = mA + nA\) is verified since both sides yield the same matrix.

Step 1: Define the Matrices and Scalars

We are given three matrices \( \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} \) and two scalars \( m \) and \( n \). The matrices \( \boldsymbol{A}, \boldsymbol{B}, \boldsymbol{C} \) are 2x2, and \( m \) and \( n \) are real numbers.

Step 2: Calculate \( (m+n) \boldsymbol{A} \)

Calculate the product of the scalar \( (m+n) \) with the matrix \( \boldsymbol{A} \).\[ (m+n) \boldsymbol{A} = (m+n) \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} (m+n)a & (m+n)b \ (m+n)c & (m+n)d \end{bmatrix} \]This involves multiplying each element of \( \boldsymbol{A} \) by the scalar sum \( (m+n) \).

Step 3: Calculate \( m \boldsymbol{A} \) and \( n \boldsymbol{A} \) Separately

Compute \( m \boldsymbol{A} \) and \( n \boldsymbol{A} \) as follows: \[ m \boldsymbol{A} = m \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} ma & mb \ mc & md \end{bmatrix} \] \[ n \boldsymbol{A} = n \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} na & nb \ nc & nd \end{bmatrix} \]Multiply each element of \( \boldsymbol{A} \) by \( m \) and \( n \) respectively.

Step 4: Add \( m \boldsymbol{A} \) and \( n \boldsymbol{A} \)

Add the matrices \( m \boldsymbol{A} \) and \( n \boldsymbol{A} \): \[ m \boldsymbol{A} + n \boldsymbol{A} = \begin{bmatrix} ma & mb \ mc & md \end{bmatrix} + \begin{bmatrix} na & nb \ nc & nd \end{bmatrix} = \begin{bmatrix} ma + na & mb + nb \ mc + nc & md + nd \end{bmatrix} \]

Key Concepts

Matrix Addition

Matrix addition is a simple and fundamental operation in linear algebra. It involves adding two matrices by adding their corresponding elements. For two matrices to be added, they must have the same dimensions. This means that if you have two matrices, each with two rows and two columns (2x2 matrices), you can add them by adding each element in one matrix to the corresponding element in the other matrix.
For example, if we have two matrices \[\begin{bmatrix}a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\] and \[\begin{bmatrix}b_{11} & b_{12} \ b_{21} & b_{22}\end{bmatrix} \], their sum will be:\[\begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22}\end{bmatrix}. \] This process simplifies to element-wise addition and is used extensively in data manipulation and transformation.

2x2 Matrices

2x2 matrices are a basic form of matrices that consist of two rows and two columns. They are often used in introductory courses to explain matrix operations because their small size makes calculations manageable while still demonstrating important concepts.
A 2x2 matrix is written as:\[\begin{bmatrix}a & b \c & d\end{bmatrix}.\] The elements \(a, b, c,\) and \(d\) can be any real numbers. These numbers fill the positions in the matrix and determine its properties and the operations that can be performed on it. For example, determinants and inverses are often calculated using these matrices due to their simplicity.

Real Numbers

Real numbers are all the numbers on the continuous number line, including all fractions, integers, and irrational numbers. In matrix operations, real numbers frequently serve as elements within matrices or as scalars for multiplication.
In the context of matrices, real numbers are used to scale matrices, whereby each entry in a matrix gets multiplied by a real number scalar. This operation is seen in problems where matrices need to be multiplied by a scalar, like when calculating \( m \textbf{A} \) or \( n \textbf{A} \). Real numbers provide the flexibility needed in scalar multiplication and matrix addition, allowing scalable vector models in various applications like data science.

Matrix Identity Verification

Matrix identity verification involves confirming that two matrix expressions are equivalent according to certain rules or properties in algebra. A common scenario is verifying identities involved in scalar multiplication and matrix addition.
For example, in our exercise, we verify that the identity \[(m+n) \textbf{A} = m \textbf{A} + n \textbf{A} \]holds true. This uses the distributive property for real numbers over matrices, showcasing how scalar multiplication distributes over addition inside a matrix. By multiplying and then adding matrices separately, you will see they yield the same results, thereby verifying the identity.
Verification is essential in confirming assumptions in matrix algebra, assisting in ensuring that equations used in more complex models and computations are correct.