Problem 31 - 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 \(n\). $$ m(A+B)=m A+m B $$

Step-by-Step Solution

Answer

The identity \(m(A+B) = mA + mB\) holds after comparing both expressions.

Step 1: Understand the Matrix Addition Property

In matrix operations, the property \((A + B)\) represents the matrix obtained by adding corresponding elements of matrices \(A\) and \(B\). For matrices of the same dimension, this operation will yield another matrix of the same dimension.

Step 2: Define the Scalar Multiplication

The operation \(m(A + B)\) involves multiplying each element of the resulting matrix \((A + B)\) by the scalar \(m\). Similarly, the expression \((mA + mB)\) involves multiplying each element of matrices \(A\) and \(B\) by \(m\) individually, yielding two separate matrices which are then added.

Step 3: Perform Element-wise Operations

Calculate \((A + B)\). \(A + B = \begin{bmatrix} a+p & b+q \ c+r & d+s \end{bmatrix}\). Now, multiply \(m\) across this matrix: \(m(A + B) = \begin{bmatrix} m(a+p) & m(b+q) \ m(c+r) & m(d+s) \end{bmatrix}\).

Step 4: Apply Scalar Multiplication Individually

Multiply matrix \(A\) by the scalar \(m\): \(mA = \begin{bmatrix} ma & mb \ mc & md \end{bmatrix}\).Multiply matrix \(B\) by the scalar \(m\): \(mB = \begin{bmatrix} mp & mq \ mr & ms \end{bmatrix}\).

Step 5: Add Scaled Matrices

Add the two scaled matrices from Step 4:\[mA + mB = \begin{bmatrix} ma+mp & mb+mq \ mc+mr & md+ms \end{bmatrix}\].

Step 6: Verify the Original Equation Holds

Compare the results of Step 3 and Step 5:For \(m(A+B)\), we have \(\begin{bmatrix} m(a+p) & m(b+q) \ m(c+r) & m(d+s) \end{bmatrix}\), which equals \( \begin{bmatrix} ma + mp & mb + mq \ mc + mr & md + ms \end{bmatrix}\).The matrices are equal, verifying the identity \(m(A+B) = mA + mB\).

Key Concepts

Matrix Addition

Matrix addition is one of the fundamental operations in matrix algebra. It involves combining two matrices of the same dimensions by adding their corresponding elements. This operation is only defined when both matrices have the same number of rows and columns. So, if you have matrices \(A\) and \(B\), each with dimensions \(m \times n\), the resulting matrix \(A + B\) will also have dimensions \(m \times n\).

• The element at the first row and first column of \(A + B\) will be the sum of the first row and first column elements of \(A\) and \(B\), respectively.
• This process applies to all elements in the matrices, ensuring that you add each element of \(A\) to the respective element of \(B\).

For example, consider the matrices \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(B = \begin{bmatrix} p & q \ r & s \end{bmatrix}\). Their sum, \(A + B\), is computed as:\[A + B = \begin{bmatrix} a+p & b+q \ c+r & d+s \end{bmatrix}.\]This operation is both commutative and associative, meaning \(A + B = B + A\) and \((A + B) + C = A + (B + C)\). These properties are crucial for establishing more complex matrix identities.

Scalar Multiplication

Scalar multiplication in matrix algebra refers to the operation of multiplying every element within a matrix by a scalar (a real number). This operation transforms the matrix by scaling each of its elements, effectively stretching or shrinking it proportionally.

• If you multiply a matrix \(A\) by a scalar \(m\), every element in \(A\) is multiplied by \(m\).
• For instance, for a matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), scalar multiplication by \(m\) results in \(mA = \begin{bmatrix} ma & mb \ mc & md \end{bmatrix}\).

Scalar multiplication is distributive over matrix addition. This means that for any two matrices \(A\) and \(B\), and a scalar \(m\), the identity \(m(A + B) = mA + mB\) holds. To prove this, start with \(A + B = \begin{bmatrix} a+p & b+q \ c+r & d+s \end{bmatrix}\), then multiply the entire resulting matrix by \(m\), leading to:\[m(A + B) = \begin{bmatrix} m(a+p) & m(b+q) \ m(c+r) & m(d+s) \end{bmatrix}.\]Separately applying scalar multiplication to \(A\) and \(B\) and then adding them as matrices gives you:\[mA + mB = \begin{bmatrix} ma & mb \ mc & md \end{bmatrix} + \begin{bmatrix} mp & mq \ mr & ms \end{bmatrix} = \begin{bmatrix} ma+mp & mb+mq \ mc+mr & md+ms \end{bmatrix}.\]Thus, the matrices are identical, proving the distributive property.

Matrix Identity Verification

Matrix identity verification involves confirming certain algebraic identities or properties hold true for matrix operations. In this context, it's crucial for ensuring that operations produce expected results.

• For the identity \(m(A + B) = mA + mB\), you need to perform calculations to compare both sides of the equation.
• Start by calculating \(A + B\) and then multiply by the scalar \(m\).
• Individually multiply each matrix, \(A\) and \(B\), by the scalar \(m\) before adding the results.

Both calculations should yield the same matrix, confirming the identity. This particular verification demonstrates the distributive property of scalar multiplication over addition. It ensures operations on matrices obey predictable rules, which is foundational for broader applications in math and science. By verifying such identities, mathematicians can confidently use matrices in complex linear equations and transformations, knowing the underlying principles remain consistent.