Problem 34 - Discrete Mathematics with Applications

Question

Let \(A, B,\) and \(C\) be any square matrices of order \(2 .\) Prove each. $$A(B+C)=A B+A C$$

Step-by-Step Solution

Answer

We proved that for any square matrices \(A\), \(B\), and \(C\) of order 2, the equation \(A(B+C) = AB + AC\) holds true. We did this by performing matrix multiplication step-by-step and comparing the resulting matrices, showing they are equal.

Step 1: Matrix Definition and Dimensions

Let \(A\), \(B\), and \(C\) be square matrices of order 2. This means that each matrix has 2 rows and 2 columns. We can represent the matrices as follows: \[A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}, C = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix}\]

Step 2: Compute B+C

From the above definitions of matrices, let's compute \(B+C\): \(B+C = \begin{pmatrix} b_{11} + c_{11} & b_{12} + c_{12} \\ b_{21} + c_{21} & b_{22} + c_{22} \end{pmatrix}\)

Step 3: Compute A(B+C)

Now let's compute \(A(B+C)\) by performing matrix multiplication: \(A(B+C) = A \begin{pmatrix} b_{11} + c_{11} & b_{12} + c_{12} \\ b_{21} + c_{21} & b_{22} + c_{22} \end{pmatrix} = \begin{pmatrix} a_{11}(b_{11} + c_{11}) + a_{12}(b_{21} + c_{21}) & a_{11}(b_{12} + c_{12}) + a_{12}(b_{22} + c_{22}) \\ a_{21}(b_{11} + c_{11}) + a_{22}(b_{21} + c_{21}) & a_{21}(b_{12} + c_{12}) + a_{22}(b_{22} + c_{22}) \end{pmatrix}\)

Step 4: Compute AB and AC

Now we will find \(AB\) and \(AC\) and add them together: \(AB = A \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix}\) \(AC = A \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix} = \begin{pmatrix} a_{11}c_{11} + a_{12}c_{21} & a_{11}c_{12} + a_{12}c_{22} \\ a_{21}c_{11} + a_{22}c_{21} & a_{21}c_{12} + a_{22}c_{22} \end{pmatrix}\) \(AB + AC = \begin{pmatrix} (a_{11}b_{11} + a_{12}b_{21}) + (a_{11}c_{11} + a_{12}c_{21}) & (a_{11}b_{12} + a_{12}b_{22}) + (a_{11}c_{12} + a_{12}c_{22}) \\ (a_{21}b_{11} + a_{22}b_{21}) + (a_{21}c_{11} + a_{22}c_{21}) & (a_{21}b_{12} + a_{22}b_{22}) + (a_{21}c_{12} + a_{22}c_{22}) \end{pmatrix}\)

Step 5: Comparing A(B+C) and AB + AC

By comparing \(A(B+C)\) and \(AB + AC\), we can see that the resulting matrices are equal: \[A(B+C)= \begin{pmatrix} a_{11}(b_{11} + c_{11}) + a_{12}(b_{21} + c_{21}) & a_{11}(b_{12} + c_{12}) + a_{12}(b_{22} + c_{22}) \\ a_{21}(b_{11} + c_{11}) + a_{22}(b_{21} + c_{21}) & a_{21}(b_{12} + c_{12}) + a_{22}(b_{22} + c_{22}) \end{pmatrix}\] \[AB + AC = \begin{pmatrix} (a_{11}b_{11} + a_{12}b_{21}) + (a_{11}c_{11} + a_{12}c_{21}) & (a_{11}b_{12} + a_{12}b_{22}) + (a_{11}c_{12} + a_{12}c_{22}) \\ (a_{21}b_{11} + a_{22}b_{21}) + (a_{21}c_{11} + a_{22}c_{21}) & (a_{21}b_{12} + a_{22}b_{22}) + (a_{21}c_{12} + a_{22}c_{22}) \end{pmatrix}\] Thus, we have proved that for any square matrices of order 2, \(A(B+C) = AB + AC\).

Key Concepts

Matrix Addition

Matrix addition is an operation where two matrices of the same dimensions are added together by adding their corresponding elements. Imagine matrices as grids of numbers. Each position in the grid matches a position in the other matrix. When adding matrices, you simply add each number in the grid to the number in the same position in the other matrix.

For instance, if you have two 2x2 matrices:- Matrix \(B = \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix} \)- Matrix \(C = \begin{pmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{pmatrix} \)

• The sum of these matrices would be \( B + C = \begin{pmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{pmatrix} \)

If matrices do not match in size, you cannot perform matrix addition. But when dimensions align, the process is straightforward!

Matrix addition is useful in various applications, such as computer graphics, where multiple transformations must be accumulated.

Matrix Multiplication

Matrix multiplication is an operation that requires a bit more work than addition. It involves multiplying rows by columns. When you multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix for the multiplication to be possible.

Consider two matrices \(A\) and \(B\), both of size 2x2, like \(A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}\) and \(B = \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix}\). The product \(AB\) is calculated as follows:

• First row, first column: \( a_{11}b_{11} + a_{12}b_{21} \)
• First row, second column: \( a_{11}b_{12} + a_{12}b_{22} \)
• Second row, first column: \( a_{21}b_{11} + a_{22}b_{21} \)
• Second row, second column: \( a_{21}b_{12} + a_{22}b_{22} \)

This results in the matrix \( AB = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix} \).

Matrix multiplication is crucial in many fields, including solving systems of linear equations and transforming geometric data.

Distributive Property of Matrices

The distributive property is a familiar concept if you've ever worked with numbers, and it applies to matrices as well. This property states that multiplying a matrix by a sum of matrices is the same as multiplying the matrix by each matrix in the sum and then adding the results.

Let's consider matrices \(A\), \(B\), and \(C\) of compatible dimensions, so we can write:

• \( A(B + C) = AB + AC \)

Using distributed multiplication, you first add matrices \(B\) and \(C\), and then you multiply the result by matrix \(A\). Alternatively, you could multiply \(A\) by each matrix \(B\) and \(C\) separately and then add the products. Both approaches yield the same result.

This property is vital for simplifying expressions in matrix algebra, making problem-solving much more manageable. It's extensively utilized in multiple applications such as network modeling, physics simulations, and computer graphics.