Problem 86 - 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. \(A(B+C)=A B+A C\) (distributive property)

Step-by-Step Solution

Answer

The distributive property holds: \(A(B+C) = AB + AC\).

Step 1: Define Matrices B and C

Define matrix B: \(B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix}\) and matrix C: \(C = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix}\). We need to compute \((B + C)\) to proceed.

Step 2: Calculate Matrix B + C

Add matrices B and C element-wise: \(B + C = \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{bmatrix}\). The operation results in the sum of the elements in their corresponding positions.

Step 3: Calculate A(B+C)

Calculate the product \(A(B+C)\) where \(A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\):\\[\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{bmatrix} = \begin{bmatrix} a_{11}(b_{11} + c_{11}) + a_{12}(b_{21} + c_{21}) & a_{11}(b_{12} + c_{12}) + a_{12}(b_{22} + c_{22}) \ a_{21}(b_{11} + c_{11}) + a_{22}(b_{21} + c_{21}) & a_{21}(b_{12} + c_{12}) + a_{22}(b_{22} + c_{22}) \end{bmatrix}\]

Step 4: Simplify A(B+C)

Rewrite the expressions by distributing the A matrix's elements: \[\begin{bmatrix} a_{11}b_{11} + a_{11}c_{11} + a_{12}b_{21} + a_{12}c_{21} & a_{11}b_{12} + a_{11}c_{12} + a_{12}b_{22} + a_{12}c_{22} \ a_{21}b_{11} + a_{21}c_{11} + a_{22}b_{21} + a_{22}c_{21} & a_{21}b_{12} + a_{21}c_{12} + a_{22}b_{22} + a_{22}c_{22} \end{bmatrix}\] This is the form of the product \(A(B+C)\).

Step 5: Calculate AB and AC Separately

Compute matrix products \(AB\) and \(AC\) separately: \(AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}\) and \(AC = \begin{bmatrix} a_{11}c_{11} + a_{12}c_{21} & a_{11}c_{12} + a_{12}c_{22} \ a_{21}c_{11} + a_{22}c_{21} & a_{21}c_{12} + a_{22}c_{22} \end{bmatrix}\).

Step 6: Verify A(B+C) = AB + AC

Add the resulting matrices from step 5: \(AB + AC = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} + a_{11}c_{11} + a_{12}c_{21} & a_{11}b_{12} + a_{12}b_{22} + a_{11}c_{12} + a_{12}c_{22} \ a_{21}b_{11} + a_{22}b_{21} + a_{21}c_{11} + a_{22}c_{21} & a_{21}b_{12} + a_{22}b_{22} + a_{21}c_{12} + a_{22}c_{22} \end{bmatrix}\). Observe that \(AB + AC = A(B+C)\) as the elements perfectly match, confirming the distributive property.

Key Concepts

Distributive Property of Matrices

In matrix algebra, the distributive property is analogous to the familiar arithmetic property. This property states that for matrices, distribution over addition is valid. For example, consider three matrices, say \(A\), \(B\), and \(C\). The distributive property tells us that multiplying \(A\) by the sum of \(B\) and \(C\) is the same as multiplying \(A\) by \(B\) and \(A\) by \(C\) separately and then adding the two products. Mathematically, this is expressed as: \[A(B+C) = AB + AC\]This property is extremely useful in simplifying complex matrix calculations and is foundational in linear algebra problems.

By understanding and utilizing the distributive property, solving matrix problems becomes much more manageable and intuitive.

Matrix Addition

Matrix addition is a straightforward concept that involves adding two matrices by combining their corresponding elements. If you have two matrices \(B\) and \(C\), addition is only possible when both matrices are of the same dimensions.

When adding two matrices:\

• Add each element in Matrix \(B\) to the corresponding element in Matrix \(C\).
• The result is a new matrix where each position is the sum of the elements from \(B\) and \(C\).

For example, consider:\[B = \begin{bmatrix}b_{11} & b_{12} \b_{21} & b_{22}\end{bmatrix},C = \begin{bmatrix}c_{11} & c_{12} \c_{21} & c_{22}\end{bmatrix}\]The sum \(B + C\) will be:\[B + C = \begin{bmatrix}b_{11} + c_{11} & b_{12} + c_{12} \b_{21} + c_{21} & b_{22} + c_{22}\end{bmatrix}\]This operation is fundamental in various applications such as solving system of equations or in computer graphics.

Matrix Multiplication

Matrix multiplication, unlike simple arithmetic multiplication, follows a specific pattern. It involves taking rows from the first matrix and columns from the second.To multiply two matrices \(A\) and \(B\), you:

• Multiply the elements of each row of \(A\) by the elements of each column in \(B\).
• Sum these products to get the elements of the resulting matrix.

Consider:\[A = \begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22}\end{bmatrix},B = \begin{bmatrix}b_{11} & b_{12} \b_{21} & b_{22}\end{bmatrix}\]The product \(AB\) is calculated as:\[AB = \begin{bmatrix}a_{11} \cdot b_{11} + a_{12} \cdot b_{21} & a_{11} \cdot b_{12} + a_{12} \cdot b_{22} \a_{21} \cdot b_{11} + a_{22} \cdot b_{21} & a_{21} \cdot b_{12} + a_{22} \cdot b_{22}\end{bmatrix}\]This systematic method ensures the transformation of input vectors from one space to another, which is invaluable in math and physics.

2x2 Matrices

2x2 matrices are among the simplest forms of matrices and are essential in introducing the basic operations of matrix mathematics.A 2x2 matrix is represented as follows:\[\begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22}\end{bmatrix}\]Here, each element \(a_{ij}\) represents the entry in the \(i\)-th row and \(j\)-th column.

2x2 matrices are particularly useful because:

• They simplify the calculation of determinants, which are used in finding inverse matrices.
• They are simple enough for manual calculations but complex enough to reveal deep mathematical properties.

Working with 2x2 matrices often serves as a stepping stone to understanding larger matrices and more complex linear algebra concepts.