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

Step-by-Step Solution

Answer

The identity \( A(B+C) = AB + AC \) holds after confirming all matrix elements match.

Step 1: Define Matrix Addition

To verify the identity \( A(B+C) = AB + AC \), start by calculating \( B+C \). To add two matrices, sum their corresponding elements. Thus, \( B+C \) is given as the matrix \[ \left[\begin{array}{cc} p + w & q + x \ r + y & s + z \end{array}\right]. \]

Step 2: Compute Matrix Multiplication for A(B+C)

Next, multiply matrix \( A \) by \( (B + C) \). Using matrix multiplication, each element in the resulting matrix is obtained by the sum of the products of rows of the first matrix with columns of the second matrix. Therefore, \( A(B+C) \) is \[ \left[\begin{array}{cc} a(p+w) + b(r+y) & a(q+x) + b(s+z) \ c(p+w) + d(r+y) & c(q+x) + d(s+z) \end{array}\right]. \]

Step 3: Calculate AB and AC Separately

Compute \( AB \) and \( AC \) individually. For \( AB \), the calculation is: \[ \left[\begin{array}{cc} ap + br & aq + bs \ cp + dr & cq + ds \end{array}\right]. \] Similarly, for \( AC \), find: \[ \left[\begin{array}{cc} aw + by & ax + bz \ cw + dy & cx + dz \end{array}\right]. \]

Step 4: Add AB and AC

Now, add the matrices \( AB \) and \( AC \). The resulting matrix is their element-wise sum: \[ \left[\begin{array}{cc} ap + br + aw + by & aq + bs + ax + bz \ cp + dr + cw + dy & cq + ds + cx + dz \end{array}\right]. \]

Step 5: Verify the Identity by Simplifying and Comparing

Compare the elements of the matrices \( A(B+C) \) and \( AB + AC \). Simplify expressions to confirm equality. For instance, the top-left element of \( A(B+C) \) is \( a(p+w) + b(r+y) \), which expands to \( ap + aw + br + by \) matching exactly the top-left element of \( AB + AC \). Similarly, verify the remaining elements match. Thus, \( A(B+C) = AB + AC \) is confirmed.

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in matrix algebra where two matrices of the same size are combined to form a new matrix. Each element in the resulting matrix is obtained by adding the respective elements from each of the matrices being added. For example, if you have two matrices:

• Matrix 1: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
• Matrix 2: \[ \begin{bmatrix} e & f \ g & h \end{bmatrix} \]

The sum of these matrices will be:\[ \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix} \]
Each position in the resulting matrix involves a simple addition of the numbers in those same positions of the original matrices.
In our example, matrices \( B \) and \( C \) are added elementwise, which is an essential step to further matrix operations.

Matrix Multiplication

Matrix multiplication involves creating a new matrix by multiplying rows of the first matrix with columns of the second. This operation is not as straightforward as matrix addition. Let's break it down to understand better.
Given matrices:

• \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \)
• \( B = \begin{bmatrix} p & q \ r & s \end{bmatrix} \)

To multiply \( A \) and \( B \), for every element \( (i, j) \) in the resulting matrix, you take a dot product of the \( i \)-th row of the first matrix and \( j \)-th column of the second matrix. For instance, the top-left element is calculated as:\[ ap + br \]This operation emphasizes the need to multiply corresponding elements and sum them for each position in the resulting matrix.
Matrix multiplication is fundamental in linear algebra due to its non-commutative nature; meaning \( AB \) is not necessarily equal to \( BA \). Understanding how to perform it correctly is key.

Identity Verification

Identity verification in this context involves proving a specific property or expression is true for given matrices. In our case, we wanted to show that \( A(B+C) = AB + AC \), which is one of the distributive properties of matrices.
To verify this identity:

• First, compute \( B+C \) by matrix addition.
• Then, perform matrix multiplication to find \( A(B+C) \).
• Separately compute each of \( AB \) and \( AC \).
• Finally, add \( AB \) and \( AC \).

A detailed examination of these calculations helps confirm that every element of \( A(B+C) \) equals the corresponding element of \( AB + AC \).
This verification is essential to understanding that matrix properties hold for all matrices and serve as tools to simplify complex computations.

Matrix Properties

Matrix properties help us in understanding the behavior of matrix operations, such as addition and multiplication. These properties are pivotal in simplifying computations in linear algebra.
Key properties include:

• Commutative property of addition: For any matrices \( A \) and \( B \), \( A+B = B+A \).
• Associative property of addition and multiplication: For addition, \( (A+B)+C = A+(B+C) \). For multiplication, \( (AB)C = A(BC) \).
• Distributive property: As illustrated in this exercise, \( A(B+C) = AB + AC \).
• Identity element for addition: Adding a zero matrix to any matrix \( A \) results in \( A \).

Understanding these properties not only aids in problem-solving but also offers insights into the linear transformations that matrices represent.
In matrix algebra, these foundational properties serve as the rules for manipulating and simplifying matrix expressions, making complex algebraic problems more manageable.