Problem 30 - College Algebra

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{rr}{-10} & {20} \\ {5} & {25}\end{array}\right], B=\left[\begin{array}{rr}{40} & {10} \\ {-20} & {30}\end{array}\right], C=\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1} \\ {1} & {0}\end{array}\right] $$ $$ A B $$

Step-by-Step Solution

Answer

The product of matrices A and B is \(\begin{bmatrix} -800 & 500 \\ -300 & 800 \end{bmatrix}\).

Step 1: Check the dimensions of matrices A and B

Matrix \(A\) is a 2x2 matrix, and matrix \(B\) is also a 2x2 matrix. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Since both matrices are 2x2, they can be multiplied together.

Step 2: Perform matrix multiplication of A and B

To multiply the matrices \(A\) and \(B\), use the formula for matrix multiplication: \[\text{(AB)}_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj}\]This means each element of the resulting matrix is the dot product of rows from matrix \(A\) and columns from matrix \(B\). Let's compute each element:- **First row, first column:** \[(-10 \times 40) + (20 \times -20) = -400 - 400 = -800\]- **First row, second column:** \[(-10 \times 10) + (20 \times 30) = -100 + 600 = 500\]- **Second row, first column:** \[(5 \times 40) + (25 \times -20) = 200 - 500 = -300\]- **Second row, second column:** \[(5 \times 10) + (25 \times 30) = 50 + 750 = 800\]

Step 3: Write the resulting matrix

The product of matrices \(A\) and \(B\) is:\[AB = \begin{bmatrix} -800 & 500 \ -300 & 800 \end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations involve performing calculations with matrices, which are arrays of numbers arranged in rows and columns.
These operations include addition, subtraction, multiplication, and finding the determinant or inverse. Each operation has specific rules and requirements.
Matrix multiplication, as illustrated in this exercise, is perhaps the most complex because it involves the dot product of rows from the first matrix with columns of the second matrix.

• To add or subtract matrices, they must have the same dimensions, meaning they must have the same number of rows and columns.
• For matrix multiplication, the number of columns in the first matrix must match the number of rows in the second.

When multiplying matrices, such as matrices A and B, it's crucial to follow the rules strictly to get the correct product. This makes understanding how matrices behave and interact essential for successfully performing these operations.

Matrix Dimensions

Matrix dimensions are pivotal in determining whether two matrices can be multiplied.
The dimensions of a matrix are given by the number of its rows followed by the number of its columns.
For instance, a 2x2 matrix has two rows and two columns.

• In this exercise, matrices A and B are both 2x2, which indicates that each has 2 rows and 2 columns.
• Matrix C would be identified with a dimension of 3x2 because it has 3 rows and 2 columns.

When multiplying matrices, the rule to check first is whether the number of columns in the first matrix is equal to the number of rows in the second matrix.
If they are, the product is possible.
Thus, for matrices A and B, the multiplication is possible and results in another 2x2 matrix.

Dot Product

The dot product is an essential component of matrix multiplication.
It involves multiplying corresponding elements and summing the products to derive a single number.
In the case of matrix operations, the dot product is used to calculate the individual elements of the resulting matrix.

• For matrix A (2x2) and matrix B (2x2), each element of the resulting matrix AB is calculated using the corresponding rows from A and columns from B.
• In our example, the top-left element of the resulting matrix is calculated as: \[(-10 \times 40) + (20 \times -20) = -800 \]
• Similarly, each subsequent element is computed by taking a row from the first matrix and a column from the second, multiplying the corresponding elements, and summing up those products.

Understanding the dot product is fundamental in matrix multiplication since it determines the value of each element in the resulting matrix.

2x2 Matrices

2x2 matrices are among the simplest forms of matrices used in mathematical operations. They consist of two rows and two columns, making them handy for basic learning and practical applications.
Working with 2x2 matrices is straightforward due to their simplicity and regularity.

• In mathematical problems, 2x2 matrices are commonly used for operations like key transformations in linear algebra, graphics, and even coding theory.
• In the matrix multiplication example, both matrices A and B are 2x2, which simplifies the process of checking whether they can be multiplied.
• The resulting matrix from the multiplication of two 2x2 matrices is also a 2x2 matrix, simplifying both execution and interpretation.

Due to their size, operations involving 2x2 matrices offer an excellent opportunity to understand matrix principles without extensive computational steps.