Problem 10 - Discrete Mathematics with Applications

Question

Let \(A=\left[\begin{array}{ccc}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{ccc}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\) . Find each. $$2 B-C$$

Step-by-Step Solution

Answer

The short answer to the question is: \(2B - C = \begin{bmatrix} 3 & -4 & 10 \\ 0 & -1 & 0\end{bmatrix}\).

Step 1: Multiply Matrix B by the Scalar 2

First, we need to multiply matrix B by the scalar value of 2. This means that each element in matrix B must be multiplied by 2. Here is the scalar multiplication calculation: Matrix B = \(\begin{bmatrix}0 & -2 & 5 \\ 0 & 0 & 1\end{bmatrix}\) 2 × Matrix B = \(\begin{bmatrix} 2(0) & 2(-2) & 2(5) \\ 2(0) & 2(0) & 2(1)\end{bmatrix}\) 2 × Matrix B = \(\begin{bmatrix} 0 & -4 & 10 \\ 0 & 0 & 2\end{bmatrix}\)

Step 2: Subtract Matrix C from Scalar-Multiplied Matrix B

Now that we have the scalar-multiplied matrix B, we need to subtract matrix C from it. Subtracting matrices simply involves subtracting the element of the second matrix from the corresponding element in the first matrix. Here is the matrix subtraction calculation: \(2B - C = \begin{bmatrix} 0 & -4 & 10 \\ 0 & 0 & 2\end{bmatrix} - \begin{bmatrix}-3 & 0 & 0 \\ 0 & 1 & 2\end{bmatrix}\) \(=\begin{bmatrix} 0-(-3) & -4-0 & 10-0 \\ 0-0 & 0-1 & 2-2\end{bmatrix}\)

Step 3: Find the Resulting Elements of the Final Matrix

With the subtraction set up element-wise, we can now perform the calculations to find the values of the resulting matrix: \(=\begin{bmatrix} 3 & -4 & 10 \\ 0 & -1 & 0\end{bmatrix}\) So, \(2B - C = \begin{bmatrix} 3 & -4 & 10 \\ 0 & -1 & 0\end{bmatrix}\).

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in matrix algebra, one that you will come across frequently when working with matrices. In essence, scalar multiplication involves taking a number, known as a scalar, and multiplying it by every entry within a matrix. This operation scales the matrix up or down depending on the value of the scalar.

For example, if we're given a matrix Matrix B and a scalar value 2, the scalar multiplication process will look like this:

• Identify the scalar value (in this case, 2).
• Multiply each element of Matrix B by 2.

The result is a new matrix where each element is twice the corresponding element in Matrix B. Precision is key here; making sure to perform the multiplication consistently across all elements is essential for accuracy in scalar multiplication.

Matrix Subtraction

Matrix subtraction is another essential operation when working with matrices. It involves taking two matrices of the same dimensions and subtracting the corresponding elements of the second matrix from the first one. This operation is only valid for matrices with the same dimensions because the subtraction must be done element-wise.

Here's how you would subtract one matrix from another:

• Ensure both matrices have the same number of rows and columns.
• Subtract each element in the second matrix from the corresponding element in the first matrix.

This operation results in a new matrix where each element reflects the difference between the elements at the same positions in the original matrices. Be cautious with negative values, as they can sometimes lead to confusion in the subtraction process.

Element-wise Matrix Operations

Element-wise matrix operations, such as the matrix subtraction outline above, require you to perform the operation on each corresponding pair of elements from two matrices. This principle doesn't just apply to subtraction; it's valid for addition, multiplication, and division, provided the latter two are defined for matrices.

Element-wise operations are straightforward: simply line up the two matrices and carry out the operation on each pair of elements.

Here are some general rules for element-wise operations:

• The matrices must be the same size.
• Perform the operation individually on each pair of corresponding elements.
• The resulting matrix will be the same size as the original matrices.

By ensuring that these conditions are met, you can perform element-wise operations consistently and correctly, leading to correct results every time you manipulate matrices.