Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{cccc}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) $$ 3 B $$

Step-by-Step Solution

Answer

The product is \( \begin{bmatrix}9 & 27 \\ 63 & 36 \\ 0 & 192\end{bmatrix} \).

Step 1: Understanding Matrix B

Matrix B is given as \(B=\begin{bmatrix}3 & 9 \ 21 & 12 \ 0 & 64\end{bmatrix}\). This matrix has 3 rows and 2 columns.

Step 2: Understanding Scalar Multiplication

Scalar multiplication involves multiplying every element of the matrix by a scalar, in this case, multiply matrix B by the scalar 3.

Step 3: Multiplicating Each Element by 3

Multiply each element of matrix B by 3. The calculations are: - First row: \(3 \times 3 = 9\), \(3 \times 9 = 27\).- Second row: \(3 \times 21 = 63\), \(3 \times 12 = 36\).- Third row: \(3 \times 0 = 0\), \(3 \times 64 = 192\).

Step 4: Writing the Resulting Matrix

Write the resulting matrix after performing the scalar multiplication:\[3B = \begin{bmatrix}9 & 27 \ 63 & 36 \ 0 & 192\end{bmatrix}\]

Key Concepts

Matrices

Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. They are an essential concept in linear algebra and are used to represent data or systems of equations. In a matrix, the horizontal lines are called rows, and the vertical ones are columns. For instance, matrix \( B \) from the example given is a 3x2 matrix, which means it has 3 rows and 2 columns.

Key features of matrices include:

• Order: Defined by the number of rows and columns it has. A matrix with \( m \) rows and \( n \) columns is referred to as an \( m \times n \) matrix.
• Elements: The individual values within the matrix. The position of each element is denoted by its row and column number.

Matrices can be manipulated through various operations, including addition, subtraction, and scalar multiplication. Scalar multiplication involves multiplying every element in a matrix by a constant, which is known as the scalar. This operation affects each entry of the matrix in the same way and is useful in various applications like graphics transformations and solving system equations.

Matrix Multiplication

Matrix multiplication involves the multiplication of two matrices to produce a third matrix. However, this operation is distinct from scalar multiplication and follows specific rules. For matrix multiplication to occur, the number of columns in the first matrix must match the number of rows in the second matrix.

Here's what you need to know about matrix multiplication:

• Compatibility: Only possible when the matrices are compatible in terms of dimensions.
• Dot Product: Each element of the resulting matrix is obtained by taking the dot product of a row from the first matrix and a column from the second matrix.
• Resulting Order: The resulting matrix's order is determined by the number of rows in the first matrix and the number of columns in the second matrix.

Matrix multiplication is not commutative, meaning that the order of matrices in multiplication matters. It's widely used in applications such as computer graphics, input-output systems, and more, where the transformation and manipulation of multi-dimensional arrays are essential.

Matrix Operations

Matrix operations encompass various methods used to manipulate matrices. These include addition, subtraction, scalar multiplication, and multiplication. Each operation has specific rules and applications that are crucial for problem-solving in mathematics and applied sciences.

Here's a closer look at common matrix operations:

• Addition and Subtraction: Only matrices of the same size can be added or subtracted. This involves adding or subtracting corresponding elements in the matrices.
• Scalar Multiplication: As demonstrated in the provided exercise, scalar multiplication involves multiplying each element of a matrix by a number (scalar), resulting in a matrix of the same size with modified values.
• Transposition: Changing the rows of a matrix to columns and vice versa. The transposed matrix is represented as \( A^T \) for a matrix \( A \).

These operations allow mathematicians and scientists to model, analyze, and solve complex systems effectively. Understanding and mastering basic matrix operations are essential skills for further study in linear algebra and real-world applications in sciences, engineering, economics, and more.