Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$\left[\begin{array}{rr}12 & -5 \\ 10 & 3\end{array}\right]-\left[\begin{array}{rr}6 & 9 \\ -2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The result is \(\begin{bmatrix} 6 & -14 \\ 12 & 3 \end{bmatrix}.\)

Step 1: Identify the Matrix Dimensions

To subtract two matrices, they must have the same dimensions. Matrix A and Matrix B are both 2x2 matrices. This means they can be subtracted from one another.

Step 2: Subtract the Matrices Element-wise

Subtract corresponding elements from both matrices. Given matrices:\[A = \begin{bmatrix} 12 & -5 \ 10 & 3 \end{bmatrix} \quad B = \begin{bmatrix} 6 & 9 \ -2 & 0 \end{bmatrix}\]Subtract each corresponding element from Matrix B from Matrix A:\[C = \begin{bmatrix} 12-6 & -5-9 \ 10-(-2) & 3-0 \end{bmatrix} = \begin{bmatrix} 6 & -14 \ 12 & 3 \end{bmatrix}\]

Step 3: Verify the Result

Double-check each step of the subtraction to ensure the results are correct. The matrix \(C = \begin{bmatrix} 6 & -14 \ 12 & 3 \end{bmatrix} \) should be confirmed.

Key Concepts

Matrix Operations

Matrix operations are fundamental tools in linear algebra, offering various ways to manipulate and use matrices for different mathematical applications. The types of matrix operations include addition, subtraction, multiplication, and finding the inverse. In each case, specific rules and conditions must be met for the operation to be carried out.

For subtraction, which is similar to addition, the basic requirement is that the matrices must have identical dimensions. This ensures a clear one-to-one correspondence between elements that allows for their subtraction. When performing operations on matrices, make sure to double-check dimensions first.

• Addition/Subtraction: Matrices must be of the same size.
• Multiplication: The number of columns in the first matrix must match the number of rows in the second.
• Inverse: Only square matrices (same number of rows and columns) may have an inverse.

Understanding these basic rules is essential to performing and verifying any matrix operation correctly.

2x2 Matrices

2x2 matrices are one of the simplest and most commonly used types of matrices in linear algebra. They consist of two rows and two columns, making them a favorite for educational examples due to their simplicity.

For example, the given exercise involves two matrices:

• Matrix A: \[\begin{bmatrix} 12 & -5 \10 & 3 \end{bmatrix} \]
• Matrix B: \[\begin{bmatrix} 6 & 9 \-2 & 0 \end{bmatrix} \]

Each element in a 2x2 matrix is defined by its position, which can be described using a simple grid layout. This makes it straightforward to match elements with their corresponding positions in a pair of matrices during operations like subtraction.

Since the layout is manageable, it's easy to visualize operations and quickly verify the accuracy of results by reviewing each computation step-by-step.

Element-wise Subtraction

Element-wise subtraction is an operation where each element in one matrix is subtracted from its corresponding element in another matrix. The matrices must have the same dimensions, ensuring that every element matches perfectly with another.

In this exercise, you subtract Matrix B from Matrix A as follows:

• Subtract the first element of B from the first element of A: \(12 - 6 = 6\).
• Subtract the second element of B from the second element of A: \(-5 - 9 = -14\).
• The third element: \(10 - (-2) = 12\), note the minus from B turns into plus.
• The fourth and final element: \(3 - 0 = 3\).

The resulting matrix C becomes: \[\begin{bmatrix} 6 & -14 \12 & 3 \end{bmatrix} \]Remember to always verify each step to confirm the accuracy, as element-wise operations can involve negative numbers, requiring careful attention to detail.