Problem 17 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform scalar multiplication. \(A=\left[\begin{array}{rr}4 & 6 \\ 13 & 12\end{array}\right], B=\left[\begin{array}{rr}3 & 9 \\ 21 & 12 \\ 0 & 64\end{array}\right], C=\left[\begin{array}{cccc}16 & 3 & 7 & 18 \\ 90 & 5 & 3 & 29\end{array}\right], D=\left[\begin{array}{rrr}18 & 12 & 13 \\ 8 & 14 & 6 \\\ 7 & 4 & 21\end{array}\right]\) $$ 100 D $$

Step-by-Step Solution

Answer

Multiply each element of matrix D by 100 for the result.

Step 1: Understand Scalar Multiplication

Scalar multiplication involves multiplying each entry of a matrix by the given scalar. Here, you have to multiply each element of matrix \(D\) by 100.

Step 2: Multiply Each Element by Scalar

Matrix \(D\) is given as \(\left[\begin{array}{rrr}18 & 12 & 13\ 8 & 14 & 6\ 7 & 4 & 21\end{array}\right]\). Multiply each element by 100. This results in:\[100 \times \begin{bmatrix} 18 & 12 & 13 \ 8 & 14 & 6 \ 7 & 4 & 21 \end{bmatrix} = \begin{bmatrix} 1800 & 1200 & 1300 \ 800 & 1400 & 600 \ 700 & 400 & 2100 \end{bmatrix}\]

Step 3: Write the Result

The resulting matrix after multiplying each element of \(D\) by 100 is \[\begin{bmatrix} 1800 & 1200 & 1300 \ 800 & 1400 & 600 \ 700 & 400 & 2100 \end{bmatrix}\].

Key Concepts

Matrices

Matrices are essential components of linear algebra.They consist of numbers or expressions arranged in rows and columns to form a rectangular grid.Each number in a matrix is called an element.
For example, Matrix \(D\) in the exercise is a 3x3 matrix, containing 3 rows and 3 columns.
Mathematical operations on matrices are often used to describe and solve linear equations, such as when dealing with systems of linear equations in algebra.

Matrix Operations

Matrix operations refer to the various calculations that can be performed on matrices, including addition, subtraction, scalar multiplication, and multiplication between two matrices. Here are some important operations:

• Addition/Subtraction: This can only be done on matrices of the same size, by adding or subtracting corresponding elements.
• Scalar Multiplication: Involves multiplying each element in a matrix by a scalar value. This operation scales the matrix but does not change its structure.
• Matrix Multiplication: This is more complex; the product of two matrices generally requires that the number of columns in the first matrix matches the number of rows in the second.

Scalar multiplication, like in the given exercise, modifies the size of each element by a constant factor.

Linear Algebra

Linear Algebra is a branch of mathematics focused on vectors, vector spaces, and linear equations.
Matrices are widely used in this field because they allow for comprehensive modeling of linear transformations and systems.
Linear algebra is vital in various real-world applications, such as physics, engineering, and computer science, often dealing with large datasets or complex systems of equations.

Matrix Multiplication

Matrix multiplication is a specific type of matrix operation that involves combining two matrices to produce a third matrix.
The key rule is that the number of columns in the first matrix must match the number of rows in the second matrix.
For matrices \(A = m \times n\) and \(B = n \times p\), the resulting matrix will have a size \(m \times p\).
Matrix multiplication is distinct from both scalar multiplication and addition, requiring care and attention to the order and dimensions of the matrices involved. Moreover, it facilitates the transformation of data and the solving of linear systems, proving useful in numerous domains of science and technology.