Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

Identity verified: \((m+n)A = mA + nA\) holds for matrix \(A\).

Step 1: Understand the Identity

We need to verify that \((m+n) A = m A + n A\) holds for a matrix \(A\). This requires distributing a scalar over matrix addition.

Step 2: Distribute Scalars

The left-hand side of the equation is \((m+n)A\), which means multiplying each element of matrix \(A\) by \((m+n)\). For matrix \(A\), this results in: \[ (m+n)A = \left[\begin{array}{cc}(m+n)a & (m+n)b \ (m+n)c & (m+n)d \end{array}\right] \]

Step 3: Calculate mA

Multiply each element of matrix \(A\) by \(m\):\[ mA = \left[\begin{array}{cc}ma & mb \ mc & md \end{array}\right] \]

Step 4: Calculate nA

Multiply each element of matrix \(A\) by \(n\):\[ nA = \left[\begin{array}{cc}na & nb \ nc & nd \end{array}\right] \]

Step 5: Add mA and nA

Add the corresponding elements of matrices \(mA\) and \(nA\):\[ mA + nA = \left[\begin{array}{cc}ma + na & mb + nb \ mc + nc & md + nd \end{array}\right] \]

Step 6: Compare Results

Compare the results from Steps 2 and 5. The elements in both resulting matrices match:\[ (m+n)A = \left[\begin{array}{cc}ma + na & mb + nb \ mc + nc & md + nd \end{array}\right] \]This confirms the identity \((m+n)A = mA + nA\).

Key Concepts

Matrix Identities

Matrix identities are fundamental expressions that reveal essential truths about matrix operations. They simplify complicated calculations and ensure we can trust our results when performing matrix computations. One key identity is the distributive property of scalar multiplication over matrix addition. This identity plays a crucial role when you need to handle matrices with real-number coefficients.

Using identities effectively requires understanding that they are based on firm mathematical principles. This identity's reliability means it can be applied consistently in various situations. By verifying identities, we leverage matrix properties like associativity and commutativity. Once applied, these identities can make complex operations more manageable and ensure our work is mathematically sound.

Matrix Addition

Matrix addition is a straightforward yet crucial operation. To add two matrices, such as matrices A and B as introduced earlier, they must have the same dimensions. This means that each matrix must have the same number of rows and columns.

To perform matrix addition, you add each corresponding element of the matrices. For example, given matrices:

• Matrix A: \(\left[\begin{array}{ll} a & b \ c & d \end{array}\right] \)
• Matrix B:\( \left[\begin{array}{ll} p & q \ r & s \end{array}\right] \)

Their sum, A + B, is formed by adding each element:\(\left[\begin{array}{ll} a+p & b+q \ c+r & d+s \end{array}\right] \).This method emphasizes that matrix addition is both intuitive and operates similarly to the familiar arithmetic addition of numbers. Crucially, the result is yet another matrix of the identical dimensions as the originals.

Scalar Multiplication Distributive Property

The scalar multiplication distributive property is pivotal when dealing with matrix operations involving real numbers. It allows us to distribute a scalar across matrix addition operations effortlessly.

To illustrate, consider the previously established identity: \((m+n) A = m A + n A\). This property tells us that when dealing with matrices and scalars, you can multiply the matrix by the sum of the scalars or distribute the multiplication to individual scalars separately first. Concretely, distributing a scalar across a matrix means multiplying every element of the matrix by that scalar. In practice:

• For a matrix \(A = \left[\begin{array}{ll} a & b \ c & d \end{array}\right] \), multiplying by \((m+n)\) gives: \(\left[\begin{array}{ll} (m+n)a & (m+n)b \ (m+n)c & (m+n)d \end{array}\right] \)
• Are equivalent to individually computed scalar products of \(mA \) and \(nA\) and summed as shown in Step 5 of the solution.
  • This confirms that operations with this property maintain correctness regardless of the order of multiplication and addition.

Using this distributive property effectively ensures computational efficiency, especially in larger, more complex matrix operations.