Problem 85 - A Graphical Approach to Precalculus with Limits

Question

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \((A B) C=A(B C)\) (associative property)

Step-by-Step Solution

Answer

The associative property holds: \((AB)C = A(BC)\) for \(2 \times 2\) matrices.

Step 1: Calculate \((AB)\)

First, calculate the product of matrices \(A\) and \(B\). We use the formula for matrix multiplication:\[(AB) = A \times B = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}\]

Step 2: Calculate \((AB)C\)

Next, multiply the result of \((AB)\) by the matrix \(C\):\[(AB)C = \begin{bmatrix} (a_{11}b_{11} + a_{12}b_{21})c_{11} + (a_{11}b_{12} + a_{12}b_{22})c_{21} & (a_{11}b_{11} + a_{12}b_{21})c_{12} + (a_{11}b_{12} + a_{12}b_{22})c_{22} \ (a_{21}b_{11} + a_{22}b_{21})c_{11} + (a_{21}b_{12} + a_{22}b_{22})c_{21} & (a_{21}b_{11} + a_{22}b_{21})c_{12} + (a_{21}b_{12} + a_{22}b_{22})c_{22} \end{bmatrix}\]

Step 3: Calculate \((BC)\)

Now, find the product of matrices \(B\) and \(C\):\[(BC) = B \times C = \begin{bmatrix} b_{11}c_{11} + b_{12}c_{21} & b_{11}c_{12} + b_{12}c_{22} \ b_{21}c_{11} + b_{22}c_{21} & b_{21}c_{12} + b_{22}c_{22} \end{bmatrix}\]

Step 4: Calculate \(A(BC)\)

Finally, multiply matrix \(A\) by the result \((BC)\):\[A(BC) = \begin{bmatrix} a_{11}(b_{11}c_{11} + b_{12}c_{21}) + a_{12}(b_{21}c_{11} + b_{22}c_{21}) & a_{11}(b_{11}c_{12} + b_{12}c_{22}) + a_{12}(b_{21}c_{12} + b_{22}c_{22}) \ a_{21}(b_{11}c_{11} + b_{12}c_{21}) + a_{22}(b_{21}c_{11} + b_{22}c_{21}) & a_{21}(b_{11}c_{12} + b_{12}c_{22}) + a_{22}(b_{21}c_{12} + b_{22}c_{22}) \end{bmatrix}\]

Step 5: Compare \((AB)C\) and \(A(BC)\)

Upon reducing and comparing both results from Step 2 and Step 4, we see:\[(AB)C = \begin{bmatrix} (a_{11}b_{11} + a_{12}b_{21})c_{11} + (a_{11}b_{12} + a_{12}b_{22})c_{21} & (a_{11}b_{11} + a_{12}b_{21})c_{12} + (a_{11}b_{12} + a_{12}b_{22})c_{22} \ (a_{21}b_{11} + a_{22}b_{21})c_{11} + (a_{21}b_{12} + a_{22}b_{22})c_{21} & (a_{21}b_{11} + a_{22}b_{21})c_{12} + (a_{21}b_{12} + a_{22}b_{22})c_{22} \end{bmatrix}\]\[A(BC) = \begin{bmatrix} a_{11}(b_{11}c_{11} + b_{12}c_{21}) + a_{12}(b_{21}c_{11} + b_{22}c_{21}) & a_{11}(b_{11}c_{12} + b_{12}c_{22}) + a_{12}(b_{21}c_{12} + b_{22}c_{22}) \ a_{21}(b_{11}c_{11} + b_{12}c_{21}) + a_{22}(b_{21}c_{11} + b_{22}c_{21}) & a_{21}(b_{11}c_{12} + b_{12}c_{22}) + a_{22}(b_{21}c_{12} + b_{22}c_{22}) \end{bmatrix}\]These two expressions are identical, thus proving the associative property: \((AB)C = A(BC)\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a process that allows us to combine two matrices to produce another matrix. If you have two matrices, say, matrix \( A \) with dimensions \( m \times n \) and matrix \( B \) with dimensions \( n \times p \), you can multiply them to get a matrix \( C \) with dimensions \( m \times p \). The key point is that the number of columns in matrix \( A \) must match the number of rows in matrix \( B \).

In terms of the operation, each element in the resulting matrix \( C \) is obtained by taking the dot product of the corresponding row of \( A \) with the column of \( B \). To dot multiply a row by a column, you multiply corresponding entries and sum them up. For example, the element in the first row and first column of \( C \) is calculated as:

• \( c_{11} = a_{11}b_{11} + a_{12}b_{21} + ext{...} + a_{1n}b_{n1} \)

Each element of the resulting matrix follows this pattern of row-column multiplication and sum. This systematic method provides a powerful tool for transforming data and composing linear transformations.

Matrix Operations

Matrix operations go beyond simple addition or multiplication. They include various calculative processes that apply specifically to matrices due to their unique structure. Some of the fundamental operations on matrices include:

• Addition: Matrices of identical dimensions can be added together. Simply add corresponding elements to form a new matrix.
• Scalar Multiplication: Multiply every element of a matrix by a constant (scalar), scaling up or down the matrix values.
• Matrix Multiplication: As detailed earlier, combines two matrices to form a new matrix based on the dot product of rows and columns.
• Transposition: Flip a matrix over its diagonal, essentially switching the matrix’s rows and columns.

Each matrix operation has particular rules and properties that must be observed to accurately perform calculations. Consistency with these rules is essential to ensuring valid mathematical results.

Properties of Matrices

Matrices have several interesting properties that form the backbone of many mathematical theories and applications. Three essential properties relevant to the associative property and matrix multiplication are:

• Associative Property: This property states that the grouping of matrices doesn’t affect the product. Mathematically, \((AB)C = A(BC)\). It's crucial for the simplification of complex expressions.
• Commutative Property: Not applicable to matrix multiplication. In general, \(AB eq BA\) even if both products are defined.
• Distributive Property: This resembles the distribution in algebra. It states that for any matrices \(A\), \(B\) and \(C\), the property \(A(B+C) = AB + AC\) holds true.

Understanding these properties of matrices enhances the ability to work with matrices effectively, ensuring your calculations reflect the true nature of linear transformations in multivariable contexts.