Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Perform each operation if possible. $$\left[\begin{array}{rr}1 & -4 \\ 2 & -3 \\ -8 & 4\end{array}\right]-\left[\begin{array}{rr}-6 & 9 \\ -2 & 5 \\ -7 & -12\end{array}\right]$$

Step-by-Step Solution

Answer

The resulting matrix is \[\begin{array}{rr} 7 & -13 \\ 4 & -8 \\ -1 & 16 \end{array}\].

Step 1: Check Matrix Dimensions

Before performing subtraction, we need to verify that both matrices have the same dimensions. The first matrix is a 3x2 matrix, and the second matrix is also a 3x2 matrix. Since both matrices have the same dimensions, subtraction is possible.

Step 2: Subtract Corresponding Elements

We subtract corresponding elements of the two matrices. This means we take the element from row 1, column 1 of the first matrix and subtract the element from row 1, column 1 of the second matrix, and so on for each element.\[\begin{array}{rr}1 & -4 \2 & -3 \-8 & 4\end{array}\] - \[\begin{array}{rr}-6 & 9 \-2 & 5 \-7 & -12\end{array}\]This results in the following calculations:\begin{align*}1 - (-6) &= 1 + 6 = 7, \-4 - 9 &= -4 - 9 = -13, \2 - (-2) &= 2 + 2 = 4, \-3 - 5 &= -3 - 5 = -8, \-8 - (-7) &= -8 + 7 = -1, \4 - (-12) &= 4 + 12 = 16.\end{align*}

Step 3: Construct the Resulting Matrix

Using the calculated values, we construct the resulting matrix from the subtraction:\[\begin{array}{rr}7 & -13 \4 & -8 \-1 & 16\end{array}\]

Key Concepts

Matrix Dimensions

Before diving into matrix operations, it is crucial to understand matrix dimensions. Think of a matrix as a rectangular grid. The dimensions of a matrix are given by the number of rows and the number of columns it has. For example, a matrix with 3 rows and 2 columns is referred to as a 3x2 matrix. To label it clearly,

• The first number tells you the count of rows.
• The second number tells you the count of columns.

This information is essential because, for operations like addition or subtraction, the matrices involved must share identical dimensions. In our case, both matrices are 3x2, allowing us to proceed with subtraction. If either matrix had a different dimension, this operation wouldn't be possible. By checking dimensions first, we ensure the operation's feasibility.

Matrix Operations

Matrix operations, such as matrix addition, subtraction, and multiplication, are fundamental in mathematics. To perform matrix subtraction, which is our focus here, both matrices must have exactly the same dimensions. Why is this so important? Because subtraction, like addition, is done element-wise. For each element in the resulting matrix, you subtract the corresponding elements from the two matrices involved. Here is a simple breakdown of the process:

• Take an element from the first matrix.
• Subtract the corresponding element from the second matrix.
• Place the result in the same position in the new matrix.

Using our example, for each position in the matrix e.g., row 1, column 1, you perform the operation: 1 from matrix one minus -6 from matrix two equals 7. This systematic subtraction continues for each element until completing the matrix. This step-by-step consistency is crucial for accurate results.

Precalculus

The concept of matrices is incredibly useful in precalculus, providing a backbone for more advanced topics. As students navigate through precalculus, they will encounter various applications for matrices. These can range from solving systems of equations to transformations in geometry.
In matrix operations specifically, understanding the preliminary steps—such as checking dimensions and conducting precise element-wise calculations—lays a solid foundation for success in more complex mathematical challenges.
With precalculus being a preparatory ground for calculus and beyond, grasping these fundamental concepts is not just beneficial but necessary. Through matrices, students also develop logical thinking and problem-solving skills, paving the way for further mathematical exploration.