Problem 33 - 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 \(n\). $$ A(B+C)=A B+A C $$

Step-by-Step Solution

Answer

The identity is verified as both sides of the equation result in the same matrix.

Step 1: Express A(B+C)

Start by finding the sum of matrices \(B\) and \(C\), which is \(B+C = \left[\begin{array}{ll}p+w & q+x\r+y & s+z\end{array}\right]\). Now, multiply this result by matrix \(A\), using matrix multiplication rules: \(A(B+C) = \left[\begin{array}{ll}a & b\c & d\end{array}\right] \left[\begin{array}{ll}p+w & q+x\r+y & s+z\end{array}\right]\).

Step 2: Calculate A(B+C)

Perform the matrix multiplication: \[ A(B+C) = \left[\begin{array}{cc} a(p+w) + b(r+y) & a(q+x) + b(s+z) \ c(p+w) + d(r+y) & c(q+x) + d(s+z) \end{array}\right] \]. This simplifies to: \[ \left[\begin{array}{cc} ap+aw+br+by & aq+ax+bs+bz \ cp+cw+dr+dy & cq+cx+ds+dz \end{array}\right] \].

Step 3: Express AB + AC Separately

First, calculate \(AB\): \[ AB = \left[\begin{array}{cc} ap+br & aq+bs \ cp+dr & cq+ds \end{array}\right] \]. Next, calculate \(AC\): \[ AC = \left[\begin{array}{cc} aw+by & ax+bz \ cw+dy & cx+dz \end{array}\right] \].

Step 4: Add AB and AC

Add the results from \(AB\) and \(AC\): \[ AB + AC = \left[\begin{array}{cc} ap+br+aw+by & aq+bs+ax+bz \ cp+dr+cw+dy & cq+ds+cx+dz \end{array}\right] \].

Step 5: Verify the Identity

Compare \(A(B+C)\) from Step 2 with \(AB + AC\) from Step 4. Both results are \[ \left[\begin{array}{cc} ap+aw+br+by & aq+ax+bs+bz \ cp+cw+dr+dy & cq+cx+ds+dz \end{array}\right] \], thereby verifying the identity \(A(B+C) = AB + AC\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental concept in matrix algebra. When you add two matrices, you add corresponding elements from each matrix to produce a new matrix. The matrices must be the same size, meaning they have the same number of rows and columns.
For matrices \( B \) and \( C \) given by \(\begin{bmatrix} p & q \ r & s \end{bmatrix}\) and \(\begin{bmatrix} w & x \ y & z \end{bmatrix}\) respectively, their addition results in a matrix \( B+C \) computed as:

• First element: \( p+w \)
• Second element: \( q+x \)
• Third element: \( r+y \)
• Fourth element: \( s+z \)

You end up with the matrix
\(\begin{bmatrix} p+w & q+x \ r+y & s+z \end{bmatrix}\).
Matrix addition holds a pivotal role as it is often a component in more complex operations such as matrix multiplication and transformations.

Distributive Property of Matrices

The distributive property of matrices is a vital rule in matrix algebra. It states that for any matrices \( A, B, \) and \( C \), the following identity holds true: \( A(B + C) = AB + AC \). This property shows how multiplication is distributed over addition.
In the given exercise, using the matrices \( A \), \( B \), and \( C \), and the identity \( A(B+C) = AB + AC \), we verified this rule. Each side of the identity simplifies to the same final matrix, demonstrating that the rule applies.
This distributive property helps in simplifying matrix expressions and is foundational in solving many algebraic problems involving matrices. By validating this property through expansion and comparison, students learn to decompose and solve matrix problems efficiently. Understanding this property is crucial as it connects the operations of addition and multiplication in matrix theory.

Matrix Algebra

Matrix Algebra is the broader study of matrices and the operations that can be performed on them. It encompasses several operations such as addition, subtraction, and multiplication, among others. In our exercise, we focused on multiplication and demonstrated the distributive property by verifying \( A(B+C) = AB + AC \).
Matrix multiplication involves calculating the dot product of rows of the first matrix with columns of the second. For instance, multiplying matrices \( A \) and \( B+C \) required summing up the products of corresponding elements in rows and columns.

• The product of the first row of \( A \) with the first column of \( B+C \) yields the element in the first row and first column of \( A(B+C) \).
• This process is repeated for each element of the resultant matrix.

Matrix Algebra is an extensive field with applications ranging from solving systems of equations to computer graphics. Mastery of matrix operations, including the distributive property and matrix addition, equips students with powerful tools for scientific computation and research in various disciplines.