Problem 31 - College Algebra with Corequisite Support

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: \(A^{2}=A \cdot A\) ) \(A=\left[\begin{array}{rr}-10 & 20 \\ 5 & 25\end{array}\right], B=\left[\begin{array}{rr}40 & 10 \\ -20 & 30\end{array}\right], C=\left[\begin{array}{rr}-1 & 0 \\ 0 & -1 \\ 1 & 0\end{array}\right]\) \(B A\)

Step-by-Step Solution

Answer

The product \(BA\) is \(\begin{bmatrix} -350 & 1050 \\ 350 & 350 \end{bmatrix}\).

Step 1: Understand the Operation

We are asked to perform matrix multiplication, specifically calculating the product \(BA\). Matrix multiplication is possible if the number of columns in the first matrix matches the number of rows in the second matrix.

Step 2: Check the Dimensions

Matrix \(B\) is of size \(2 \times 2\) (2 rows and 2 columns), and matrix \(A\) is also \(2 \times 2\). Since both matrices are of size \(2 \times 2\), the multiplication \(BA\) is possible.

Step 3: Perform the Matrix Multiplication

To compute \(BA\), multiply each element of the rows of matrix \(B\) by each element of the columns of matrix \(A\) and sum them. The resulting matrix will also be \(2 \times 2\).- The element at position \((1,1)\) in \(BA\) is computed as \( (40)(-10) + (10)(5) = -400 + 50 = -350\).- The element at position \((1,2)\) in \(BA\) is \( (40)(20) + (10)(25) = 800 + 250 = 1050\).- The element at position \((2,1)\) in \(BA\) is \( (-20)(-10) + (30)(5) = 200 + 150 = 350\).- The element at position \((2,2)\) in \(BA\) is \( (-20)(20) + (30)(25) = -400 + 750 = 350\).

Step 4: Write the Resultant Matrix

After performing the multiplication, we get the resultant matrix:\[BA = \begin{bmatrix} -350 & 1050 \ 350 & 350 \end{bmatrix}\]

Key Concepts

Matrices

Matrices are fundamental tools in mathematics, often represented as rectangular arrays of numbers, symbols, or expressions. They consist of rows and columns and are typically enclosed by square brackets. Here are a few key points:

• Matrices can have different shapes and sizes, depending on the number of rows and columns they contain.
• Each entry in a matrix is known as an element, and the position of an element is usually noted by its row and column numbers.
• Matrices are widely used in various fields such as physics, computer graphics, and statistics to represent complex data and solve systems of equations.

Understanding matrices is essential before delving into more complex operations such as matrix multiplication.

Matrix Dimensions

Matrix dimensions describe the size of a matrix, defined by two numbers: the number of rows and the number of columns. These numbers are expressed in the format 'rows × columns'. Here are some useful points:

• A matrix with the same number of rows and columns is called a square matrix.
• The dimensions are critical in determining whether certain matrix operations, like multiplication, are possible.
• Knowing the dimensions helps you understand how matrices can be added, subtracted, or multiplied.

For instance, in our problem, both matrices A and B are 2 × 2, meaning each has 2 rows and 2 columns. This makes them both square matrices.

Matrix Operations

Matrix operations include addition, subtraction, multiplication, and other complex calculations. Let's look at some key matrix operations:

• Addition and Subtraction: These operations can only be performed on matrices with identical dimensions.
• Scalar Multiplication: Each element of the matrix is multiplied by a constant (a scalar).
• Matrix Multiplication: More complex than addition and subtraction, it involves multiplying rows of the first matrix by columns of the second.

In the exercise, we focus on matrix multiplication, where specific rules for dimensions must be followed.

Product of Matrices

The product of matrices, or matrix multiplication, is more intricate than adding or subtracting matrices. Here is how it works:

• Matrix multiplication requires that the number of columns in the first matrix matches the number of rows in the second matrix.
• The resulting matrix's dimensions come from the outer dimensions of the two matrices being multiplied.
• Each element in the resulting matrix is computed by summing the products of corresponding elements from the rows of the first matrix and the columns of the second.

For our exercise, computing \(BA\), since both matrices B and A are 2 × 2, the resulting matrix is also 2 × 2. Each element in \(BA\) is obtained by following the multiplication and addition steps for each row and column pair.