Problem 38 - College Algebra

Question

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows If an operation is not defined, state the reason. $$A=\left[\begin{array}{rr}4 & 0 \\\\-3 & 5 \\\0 & 1\end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\\\-2 & -2\end{array}\right] \quad C=\left[\begin{array}{rr}1 & -1 \\\\-1 & 1\end{array}\right]$$ $$5 C-2 B$$

Step-by-Step Solution

Answer

The result of \(5C - 2B\) is \( \left[\begin{array}{rr}-5 & -3\\-1 & 1\end{array}\right] \).

Step 1: Matrix Scalar Multiplication

The scalar multiplication of the matrix means multiplying every element of the matrix with the scalar. So, multiply matrix \( C = \left[\begin{array}{rr}1 & -1\\-1 & 1\end{array}\right] \) by 5 and matrix \( B = \left[\begin{array}{rr}5 & 1\\-2 & -2\end{array}\right] \) by 2.

Step 2: Compute the results of scalar multiplication

Obtain two new matrices: For 5C, the result would be \( \left[\begin{array}{rr}5 & -5\\-5 & 5\end{array}\right] \) and for 2B, the result would be \( \left[\begin{array}{rr}10 & 2\\-4 & -4\end{array}\right] \).

Step 3: Matrix Subtraction

Matrix subtraction is calculated elementwise. Subtract each element in the 2B matrix from the corresponding element in 5C matrix.

Step 4: Compute the Subtraction

After subtracting, the resulting matrix is \( \left[\begin{array}{rr}-5 & -3\\-1 & 1\end{array}\right] \).

Key Concepts

Matrix Scalar Multiplication

Matrix scalar multiplication is a fundamental operation in matrix algebra where a matrix is multiplied by a scalar (a single number). To perform this operation, each element of the matrix is multiplied by the scalar. This changes the values within the matrix but maintains its dimensions.

For example, in the exercise given, we had to multiply matrix \( C = \left[\begin{array}{rr}1 & -1\-1 & 1\end{array}\right] \) by the scalar 5. This scalar multiplication results in the new matrix \( 5C = \left[\begin{array}{rr}5 & -5\-5 & 5\end{array}\right] \).
Similarly, multiplying matrix \( B = \left[\begin{array}{rr}5 & 1\-2 & -2\end{array}\right] \) by the scalar 2 results in \( 2B = \left[\begin{array}{rr}10 & 2\-4 & -4\end{array}\right] \).

Key points to remember:

• Multiply each element of the matrix by the scalar.
• The dimensions of the resulting matrix remain unchanged.
• This operation is distributive over matrix addition.

Matrix Subtraction

Matrix subtraction involves subtracting one matrix from another, assuming they have the same dimensions. This operation is conducted elementwise, meaning each corresponding element in the matrices is subtracted individually.

In our exercise, after performing scalar multiplication on matrices \( C \) and \( B \), we obtained matrices \( 5C \) and \( 2B \).
To find \( 5C - 2B \), we subtract corresponding elements of \( 2B \) from \( 5C \):

• Top-left elements: \( 5 - 10 = -5 \)
• Top-right elements: \( -5 - 2 = -7 \)
• Bottom-left elements: \( -5 - (-4) = -1 \)
• Bottom-right elements: \( 5 - (-4) = 9 \)

This results in the matrix \( \left[\begin{array}{rr}-5 & -7\-1 & 9\end{array}\right] \).

Important considerations:

• Matrices must have the same dimensions for subtraction to be defined.
• Subtraction is performed elementwise.

Matrix Algebra

Matrix algebra encompasses various operations involving matrices. These include addition, subtraction, multiplication (both scalar and matrix), and operations like finding the transpose and determinant.

The exercise demonstrated matrix scalar multiplication and subtraction, two vital operations in matrix algebra.
In the broader context:

• Addition/Subtraction: Perform these operations elementwise and ensure matrices have matching dimensions.
• Scalar Multiplication: Multiply each element in a matrix by a scalar.
• Matrix Multiplication: More complex, involving the dot product of rows and columns.

Matrix algebra is essential for solving numerous mathematical problems, from system equations to transformations in computer graphics.

Elementwise Operations

Elementwise operations mean performing calculations on corresponding elements across two matrices or within a matrix itself. In matrix algebra, these operations are straightforward but powerful, allowing us to manipulate and analyze matrices easily.

In the exercise, once the scalar multiplications were complete, we conducted an elementwise operation during the matrix subtraction part.
Each element in the result matrix comes from subtracting the element in matrix \( 2B \) from the corresponding element in matrix \( 5C \). This highlights how matrix operations can be broken down into smaller, manageable tasks.

Key aspects of elementwise operations:

• Requires matrices to have the same dimensions.
• Allows flexible, element-specific modifications.
• Can be applied to addition, subtraction, multiplication, and division.

Understanding these operations makes working with matrices less intimidating and more intuitive.