MTH120 College Algebra Chapter 7.5

 7.5 Matrices and Matrix Operations

Matrices are a fundamental concept in linear algebra and various areas of mathematics, science, and engineering. They provide a way to organize and manipulate data in a structured format. In this section, we'll explore matrices and their operations:

**1. What is a Matrix?

• A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns.
• Each element in a matrix is called an entry or a component. Entries are often denoted using subscripts, such as ${�}_{��}$ to represent the element in the $�$-th row and $�$-th column.
• Matrices are used to represent data, equations, transformations, and more.

**2. Types of Matrices:

• Row Matrix: A matrix with only one row is called a row matrix or row vector.
• Column Matrix: A matrix with only one column is called a column matrix or column vector.
• Square Matrix: A matrix with the same number of rows and columns.
• Scalar Matrix (or Identity Matrix): A square matrix with ones on the main diagonal and zeros elsewhere.
• Zero Matrix: A matrix where all entries are zero.

**3. Matrix Operations:

• Matrix Addition: You can add two matrices of the same size by adding corresponding entries.
• Matrix Subtraction: Similar to addition, subtract matrices by subtracting corresponding entries.
• Scalar Multiplication: Multiply a matrix by a scalar (a single number) by multiplying each entry by the scalar.
• Matrix Multiplication: To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The result is a new matrix.
• Matrix Transposition: Obtain the transpose of a matrix by switching its rows and columns.
• Matrix Inversion: Inverse of a square matrix (if it exists) is a matrix that, when multiplied with the original matrix, results in the identity matrix.

**4. Applications of Matrices:

• Matrices are widely used in solving systems of linear equations.
• They are used in computer graphics for transformations and rendering.
• In physics and engineering, matrices describe linear transformations and systems of equations.
• In statistics, matrices are used for data analysis and multivariate statistics.

**5. Matrix Notation:

• Matrices are often represented using capital letters (e.g., $�$) or bold symbols.
• Elements of a matrix are denoted as ${�}_{��}$, where $�$ is the row number and $�$ is the column number.
• The size of a matrix is denoted as $�\times �$, where $�$ is the number of rows and $�$ is the number of columns.

Matrix operations play a crucial role in various mathematical and scientific fields, making them a fundamental concept to understand. They provide a powerful tool for representing and manipulating data and equations in a structured format.

Finding the sum and difference of two matrices is a straightforward process, as long as the matrices have the same dimensions (the same number of rows and columns). Here's how you can find the sum and difference of two matrices, along with an example for each:

Matrix Addition (Sum): To add two matrices $�$ and $�$, which must have the same dimensions ($�\times �$), simply add their corresponding elements together.

If $�=[{�}_{��}]$ and $�=[{�}_{��}]$, then the sum $�=�+�$ is given by: ${�}_{��}={�}_{��}+{�}_{��}$

Matrix Subtraction (Difference): To subtract one matrix $�$ from another matrix $�$, which must have the same dimensions ($�\times �$), subtract their corresponding elements.

If $�=[{�}_{��}]$ and $�=[{�}_{��}]$, then the difference $�=�-�$ is given by: ${�}_{��}={�}_{��}-{�}_{��}$

Here are examples of both operations:

Matrix Addition (Sum) Example: Let's say you have two matrices $�$ and $�$ with the following entries:

$�=\left[\begin{array}{ccc}2& 4& 6\\ 1& 3& 5\end{array}\right]$

$�=\left[\begin{array}{ccc}1& 1& 1\\ 2& 2& 2\end{array}\right]$

To find the sum $�=�+�$, simply add the corresponding elements:

$�=\left[\begin{array}{ccc}2+1& 4+1& 6+1\\ 1+2& 3+2& 5+2\end{array}\right]=\left[\begin{array}{ccc}3& 5& 7\\ 3& 5& 7\end{array}\right]$

So, the sum of matrices $�$ and $�$ is $�$.

Matrix Subtraction (Difference) Example: Consider two matrices $�$ and $�$ with the following entries:

$�=\left[\begin{array}{cc}5& 3\\ 2& 1\end{array}\right]$

$�=\left[\begin{array}{cc}2& 1\\ 1& 0\end{array}\right]$

To find the difference $�=�-�$, simply subtract the corresponding elements:

$�=\left[\begin{array}{cc}5-2& 3-1\\ 2-1& 1-0\end{array}\right]=\left[\begin{array}{cc}3& 2\\ 1& 1\end{array}\right]$

So, the difference of matrices $�$ and $�$ is $�$.

Matrix addition and subtraction are essential operations in linear algebra, and they often arise in various mathematical and scientific applications. Make sure that the matrices you are adding or subtracting have the same dimensions to perform these operations.

Adding and Subtracting Matrices:

When adding or subtracting matrices, it's crucial that the matrices have the same dimensions. If $�$ and $�$ are matrices of the same size ($�\times �$), the sum $�+�$ and the difference $�-�$ are computed by adding or subtracting corresponding elements.

Matrix Addition (Sum): $�=�+�$ ${�}_{��}={�}_{��}+{�}_{��}$

Matrix Subtraction (Difference): $�=�-�$ ${�}_{��}={�}_{��}-{�}_{��}$

Let's go through examples of both operations:

Matrix Addition Example: Let $�=\left[\begin{array}{cc}2& 4\\ 1& 3\end{array}\right]$ and $�=\left[\begin{array}{cc}1& 1\\ 2& 2\end{array}\right]$

The sum $�=�+�$ is calculated as: $�=\left[\begin{array}{cc}2+1& 4+1\\ 1+2& 3+2\end{array}\right]=\left[\begin{array}{cc}3& 5\\ 3& 5\end{array}\right]$

So, $�$ is a matrix of the same size as $�$ and $�$, where each element is the sum of the corresponding elements of $�$ and $�$.

Matrix Subtraction Example: Let $�=\left[\begin{array}{cc}5& 3\\ 2& 1\end{array}\right]$ and $�=\left[\begin{array}{cc}2& 1\\ 1& 0\end{array}\right]$

The difference $�=�-�$ is calculated as: $�=\left[\begin{array}{cc}5-2& 3-1\\ 2-1& 1-0\end{array}\right]=\left[\begin{array}{cc}3& 2\\ 1& 1\end{array}\right]$

So, $�$ is a matrix of the same size as $�$ and $�$, where each element is the difference of the corresponding elements of $�$ and $�$.

These operations are straightforward as long as the matrices involved have the same dimensions. If the matrices have different dimensions, addition and subtraction are not defined. Always check the dimensions before performing these operations.

Finding scalar multiples of a matrix involves multiplying each element (entry) in the matrix by a scalar (a single number). This operation results in a new matrix of the same size as the original matrix, where each element is the product of the corresponding element in the original matrix and the scalar.

Here's how to find scalar multiples of a matrix, along with examples:

Scalar Multiplication of a Matrix: Let $�$ be a matrix and $�$ be a scalar. To find the scalar multiple $��$, multiply each element of matrix $�$ by the scalar $�$.

For a matrix $�=[{�}_{��}]$ and scalar $�$, the scalar multiple $��$ is given by: $��=[�{�}_{��}]$

Examples:

Example 1: Let's say you have the following matrix $�$ and want to find $2�$, which is the scalar multiple of $�$ by 2:

$�=\left[\begin{array}{cc}3& -1\\ 4& 2\end{array}\right]$

To find $2�$, simply multiply each element of $�$ by 2:

$2�=\left[\begin{array}{cc}2\cdot 3& 2\cdot (-1)\\ 2\cdot 4& 2\cdot 2\end{array}\right]=\left[\begin{array}{cc}6& -2\\ 8& 4\end{array}\right]$

So, $2�$ is a scalar multiple of $�$, and each element in the resulting matrix is twice the corresponding element in matrix $�$.

Example 2: Now, let's find $(-1.5)�$, which is the scalar multiple of $�$ by -1.5:

$�=\left[\begin{array}{cc}7& 1\\ -3& 5\end{array}\right]$

To find $(-1.5)�$, multiply each element of $�$ by -1.5:

$-1.5�=\left[\begin{array}{cc}-1.5\cdot 7& -1.5\cdot 1\\ -1.5\cdot (-3)& -1.5\cdot 5\end{array}\right]=\left[\begin{array}{cc}-10.5& -1.5\\ 4.5& -7.5\end{array}\right]$

So, $(-1.5)�$ is a scalar multiple of $�$, and each element in the resulting matrix is -1.5 times the corresponding element in matrix $�$.

Scalar multiplication is a fundamental operation in linear algebra and is used in various applications, such as scaling transformations in computer graphics and physics.

Here are a couple of examples of scalar multiplication with matrices:

Example 1: Scalar Multiplication of a Matrix by 2

Let's say you have the following matrix $�$ and you want to find $2�$, which is the result of multiplying $�$ by the scalar 2:

$�=\left[\begin{array}{cc}3& -1\\ 4& 2\end{array}\right]$

To find $2�$, simply multiply each element of $�$ by 2:

$2�=\left[\begin{array}{cc}2\cdot 3& 2\cdot (-1)\\ 2\cdot 4& 2\cdot 2\end{array}\right]=\left[\begin{array}{cc}6& -2\\ 8& 4\end{array}\right]$

So, $2�$ is a scalar multiple of $�$, and each element in the resulting matrix is twice the corresponding element in matrix $�$.

Example 2: Scalar Multiplication of a Matrix by -0.5

Now, let's find $(-0.5)�$, which is the result of multiplying $�$ by the scalar -0.5:

$�=\left[\begin{array}{cc}7& 1\\ -3& 5\end{array}\right]$

To find $(-0.5)�$, multiply each element of $�$ by -0.5:

$-0.5�=\left[\begin{array}{cc}-0.5\cdot 7& -0.5\cdot 1\\ -0.5\cdot (-3)& -0.5\cdot 5\end{array}\right]=\left[\begin{array}{cc}-3.5& -0.5\\ 1.5& -2.5\end{array}\right]$

So, $(-0.5)�$ is a scalar multiple of $�$, and each element in the resulting matrix is -0.5 times the corresponding element in matrix $�$.

Scalar multiplication is a basic operation in linear algebra and is used in various applications, such as scaling transformations in computer graphics, physics, and engineering. It allows you to stretch or shrink a matrix while preserving its structure.

Finding the product of two matrices involves multiplying the rows of the first matrix by the columns of the second matrix. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. The resulting matrix will have dimensions equal to the number of rows in the first matrix by the number of columns in the second matrix.

Here are a couple of examples of matrix multiplication:

Example 1: Matrix Multiplication

Let's say you have two matrices $�$ and $�$ and you want to find their product $�=��$:

Matrix $�$ is a 2x3 matrix: $�=\left[\begin{array}{ccc}2& 3& 1\\ -1& 0& 2\end{array}\right]$

Matrix $�$ is a 3x2 matrix: $�=\left[\begin{array}{cc}1& -2\\ 4& 1\\ 0& 3\end{array}\right]$

To find $�=��$, perform the matrix multiplication by taking the dot product of rows in $�$ and columns in $�$:

$�=\left[\begin{array}{ccc}2& 3& 1\\ -1& 0& 2\end{array}\right]\left[\begin{array}{cc}1& -2\\ 4& 1\\ 0& 3\end{array}\right]=\left[\begin{array}{cc}2\cdot 1+3\cdot 4+1\cdot 0& 2\cdot (-2)+3\cdot 1+1\cdot 3\\ -1\cdot 1+0\cdot 4+2\cdot 0& -1\cdot (-2)+0\cdot 1+2\cdot 3\end{array}\right]=\left[\begin{array}{cc}14& 5\\ -1& 8\end{array}\right]$

So, the product of matrices $�$ and $�$ is $�=\left[\begin{array}{cc}14& 5\\ -1& 8\end{array}\right]$.

Example 2: Matrix Multiplication

Now, let's find the product of two matrices $�$ and $�$:

Matrix $�$ is a 2x2 matrix: $�=\left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right]$

Matrix $�$ is a 2x2 matrix: $�=\left[\begin{array}{cc}4& 0\\ 2& -1\end{array}\right]$

To find $�=��$, perform the matrix multiplication:

$�=\left[\begin{array}{cc}2& 1\\ -1& 3\end{array}\right]\left[\begin{array}{cc}4& 0\\ 2& -1\end{array}\right]=\left[\begin{array}{cc}2\cdot 4+1\cdot 2& 2\cdot 0+1\cdot (-1)\\ -1\cdot 4+3\cdot 2& -1\cdot 0+3\cdot (-1)\end{array}\right]=\left[\begin{array}{cc}10& -1\\ 2& -3\end{array}\right]$

So, the product of matrices $�$ and $�$ is $�=\left[\begin{array}{cc}10& -1\\ 2& -3\end{array}\right]$.

Matrix multiplication is a fundamental operation in linear algebra and is used in various applications, such as solving systems of linear equations and transformations in computer graphics and physics. It's important to ensure that the dimensions of the matrices are compatible for multiplication.

I can help you understand how to evaluate matrix operations using a calculator. While I can't physically perform calculations, I can guide you through the process. To evaluate matrix operations using a calculator, follow these steps:

1. Turn on Your Calculator: Make sure your calculator is powered on and ready for use.

2. Enter the Matrices: Input the matrices you want to perform operations on. Most calculators have a specific key or menu for matrices. Enter the values of each matrix row by row and use the appropriate separator (usually a comma) to distinguish elements.

3. Select the Operation: Depending on your calculator model, there might be dedicated keys or a menu for matrix operations. Choose the operation you want to perform (e.g., addition, subtraction, multiplication).

4. Specify the Matrices: Indicate which matrices you want to operate on. You might need to assign names or variables to the matrices for easier input. For example, you could label matrix $�$ as "A" and matrix $�$ as "B."

5. Perform the Operation: Execute the chosen operation. The calculator will compute the result based on the selected matrices and operation.

6. View the Result: The calculator will display the result on the screen. If the result is a matrix, you'll see the new matrix with its elements.

Keep in mind that the specific steps and menu options may vary depending on the brand and model of your calculator. It's essential to consult your calculator's manual or documentation to understand how to perform matrix operations correctly.

Here's an example of how to perform matrix multiplication on a typical scientific calculator:

Let's say you have two matrices:

Matrix $�$: $�=\left[\begin{array}{cc}2& 3\\ 1& 4\end{array}\right]$

Matrix $�$: $�=\left[\begin{array}{cc}5& 1\\ 0& 2\end{array}\right]$

You want to calculate the product $�=��$. To do this on a calculator:

1. Enter matrix $�$ and matrix (B).
2. Select the matrix multiplication operation (usually a "×" or "*" symbol on the calculator).
3. Specify matrices $�$ and (B) as the operands.
4. Execute the operation.

The calculator will display the result matrix $�$, which in this case will be:

$�=\left[\begin{array}{cc}10& 8\\ 5& 9\end{array}\right]$

Please consult your specific calculator's documentation for detailed instructions on matrix operations, as different models may have variations in their user interfaces and operation processes.

Let's perform matrix addition and subtraction using the provided matrices. Remember that to add or subtract matrices, they must have the same dimensions.

Given matrices:

A = $\left[\begin{array}{cc}10& 3\\ 7& 5\end{array}\right]$

B = $\left[\begin{array}{cc}2& 2\\ 2& 4\\ 1& 6\end{array}\right]$

C = $\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$

D = $\left[\begin{array}{ccc}10& 7& 5\\ 1& 4& 2\\ 6& 1& 4\end{array}\right]$

E = $\left[\begin{array}{ccc}6& 1& 4\\ 1& 2& 5\end{array}\right]$

F = $\left[\begin{array}{ccc}0& 7& 8\\ 1& 5& 9\\ 1& 7& 4\end{array}\right]$

Now, let's perform the operations:

6. A + B:
   Matrix A is a 2x2 matrix, and matrix B is a 3x2 matrix. The operation is undefined because the matrices have different numbers of rows.

7. C + D:
   Matrix C is a 2x3 matrix, and matrix D is a 3x3 matrix. The operation is undefined because the matrices have different numbers of columns.

8. A + C:
   Matrix A is a 2x2 matrix, and matrix C is a 2x3 matrix. The operation is undefined because the matrices have different dimensions.

9. B - E:
   Matrix B is a 3x2 matrix, and matrix E is a 2x3 matrix. The operation is undefined because the matrices have different dimensions.

10. C + F:
    Matrix C is a 2x3 matrix, and matrix F is a 3x3 matrix. The operation is undefined because the matrices have different dimensions.

11. D - B:
    Matrix D is a 3x3 matrix, and matrix B is a 3x2 matrix. The operation is defined, and we can perform the subtraction.

Let's calculate D - B:

$�-�=\left[\begin{array}{ccc}10& 7& 5\\ 1& 4& 2\\ 6& 1& 4\end{array}\right]-\left[\begin{array}{cc}2& 2\\ 2& 4\\ 1& 6\end{array}\right]$

To subtract the matrices, subtract corresponding elements:

$�-�=\left[\begin{array}{cc}10-2& 7-2\\ 1-2& 4-4\\ 6-1& 1-6\end{array}\right]=\left[\begin{array}{cc}8& 5\\ -1& 0\\ 5& -5\end{array}\right]$

So, the result of D - B is:

$�-�=\left[\begin{array}{cc}8& 5\\ -1& 0\\ 5& -5\end{array}\right]$

The operations that were undefined are A + B, C + D, A + C, B - E, and C + F, and the result of D - B is $\left[\begin{array}{cc}8& 5\\ -1& 0\\ 5& -5\end{array}\right]$.