Problem 43 - Algebra for College Students

Question

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \((A+B) C=A C+B C\).

Step-by-Step Solution

Answer

\((A + B)C = AC + BC\) is true as demonstrated by identical element-wise computation.

Step 1: Expand (A + B)C

To show that \((A + B)C = AC + BC\), start by calculating \((A + B)C\). The matrix \(A + B\) is the sum of matrices \(A\) and \(B\), which results in \(\left[\begin{array}{cc} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{array}\right]\). Now multiply this matrix by \(C\), which is \(\left[\begin{array}{cc} c_{11} & c_{12} \ c_{21} & c_{22} \end{array}\right]\).

Step 2: Calculate Each Element in the Product

Now calculate each element in the resulting matrix from \((A + B)C\). For the first row and first column element:\((a_{11} + b_{11})c_{11} + (a_{12} + b_{12})c_{21}\). For the first row, second column:\((a_{11} + b_{11})c_{12} + (a_{12} + b_{12})c_{22}\). For these, we do a similar calculation for the second row elements.

Step 3: Expand AC and BC Separately

Expand \(AC\), which gives the elements:- First row: \(a_{11}c_{11} + a_{12}c_{21}, \; a_{11}c_{12} + a_{12}c_{22}\)- Second row: \(a_{21}c_{11} + a_{22}c_{21}, \; a_{21}c_{12} + a_{22}c_{22}\)Similarly, expand \(BC\), which gives:- First row: \(b_{11}c_{11} + b_{12}c_{21}, \; b_{11}c_{12} + b_{12}c_{22}\)- Second row: \(b_{21}c_{11} + b_{22}c_{21}, \; b_{21}c_{12} + b_{22}c_{22}\).

Step 4: Add the Products AC and BC

Add the results from \(AC\) and \(BC\) together:- First row elements: - First position: \(a_{11}c_{11} + a_{12}c_{21} + b_{11}c_{11} + b_{12}c_{21}\) - Second position: \(a_{11}c_{12} + a_{12}c_{22} + b_{11}c_{12} + b_{12}c_{22}\)- Second row elements: - First position: \(a_{21}c_{11} + a_{22}c_{21} + b_{21}c_{11} + b_{22}c_{21}\) - Second position: \(a_{21}c_{12} + a_{22}c_{22} + b_{21}c_{12} + b_{22}c_{22}\).

Step 5: Conclusion

Now compare each element from \((A + B)C\) and \(AC + BC\). They are identical:- First row and first column: \((a_{11} + b_{11})c_{11} + (a_{12} + b_{12})c_{21} = a_{11}c_{11} + a_{12}c_{21} + b_{11}c_{11} + b_{12}c_{21}\)- Perform similar checks for all other positions.Thus, we confirm that \((A + B)C = AC + BC\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in matrix algebra where you add two matrices of the same dimension by adding their corresponding elements. When you look at matrices, you can think of them as grids of numbers. This makes adding them pretty straightforward! If matrix \(A\) is a \(2 \times 2\) matrix and looks like \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \), and matrix \(B\) is also a \(2 \times 2\) matrix, say \( \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} \), you add \(A\) and \(B\) by adding up the numbers in corresponding positions. So, you would end up with \( \begin{bmatrix} 1+5 & 2+6 \ 3+7 & 4+8 \end{bmatrix} \), which simplifies to \( \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix} \).

• The key rule in matrix addition is that both matrices must be of the same size.
• Moreover, the sum of matrices results in another matrix of the same dimensions.

This operation is quite simple, but understanding it is crucial as it forms a basis for more complex operations like matrix multiplication.

Distributive Property

In mathematics, the distributive property is a law that states that multiplying a number by a sum is the same as doing each multiplication separately. This property also applies to matrices! When we're dealing with matrices, the distributive property helps us simplify expressions and equations.
Consider the original exercise that wants us to show that \((A + B)C = AC + BC\). The distributive property tells us when we multiply a sum like \((A + B)\) by another matrix \(C\), it's the same as multiplying each addend separately by \(C\) and then adding up the results.

• In our step-by-step solution, we first expanded \((A + B)C\) to achieve a new matrix.
• Then, we expanded \(AC\) and \(BC\) separately.
• Finally, adding the results from \(AC\) and \(BC\) should yield the same outcome as \((A+B)C\).

This way, the distributive property helps to ensure that our calculations hold true across different forms of the equation.

Matrix Algebra

Matrix algebra is a powerful branch of mathematics that allows for managing and solving linear equations using matrices. Matrices can represent numerous mathematical objects and systems efficiently. We use operations like addition, multiplication, and more complex functions to work with these matrices in matrix algebra.
One important operation in matrix algebra is matrix multiplication. This operation is not the same as multiplying numbers! Instead of multiplying cell by cell, each element of the resulting matrix is the sum of the products of each row from the first matrix and each column from the second matrix. It sounds tricky, but with practice, it becomes routine.
To multiply a matrix by another, follow these steps:

• Ensure the number of columns in the first matrix equals the number of rows in the second matrix.
• The product will have as many rows as the first matrix and as many columns as the second matrix.
• Compute each element by summing the products of corresponding entries from the row of the first matrix and the column of the second.

Matrix algebra also involves using properties like the distributive, associative, and commutative properties to simplify and transform expressions, making it a valuable tool in fields such as physics and engineering.

Linear Algebra

Linear algebra is a major area of mathematics that focuses on vectors, vector spaces, and linear transformations. At its core, linear algebra deals with systems of linear equations and finding solutions that describe linear relationships. It's a foundational concept that underscores many applications in engineering, physics, mathematics, and computer science.
In linear algebra, matrices are used as a tool to represent and solve systems of linear equations concisely. For instance, a system of equations can be represented as a matrix equation, simplifying the process of solving multiple equations at once.

• Consider a system of equations that can be represented as \(AX = B\), where \(A\) is a matrix of coefficients, \(X\) is a column matrix of variables, and \(B\) is a column matrix of constants.
• The aim is to find \(X\) such that the equation holds true.
• Linear algebra provides the tools to do this through techniques like Gaussian elimination, matrix inversion, and using eigenvalues and eigenvectors.

Understanding linear algebra opens doors to more advanced mathematical concepts and allows for practical applications in data science, machine learning, and more.