Problem 21 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{rrr}{-9} & {2} & {-7} \\ {8} & {10} & {3} \\ {-7} & {4} & {15}\end{array}\right]-\left[\begin{array}{rrr}{-1} & {3} & {6} \\ {-7} & {-3} & {5} \\ {2} & {11} & {-4}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \( \begin{array}{rrr} -8 & -1 & -13 \\ 15 & 13 & -2 \\ -9 & -7 & 19 \\ \end{array} \).

Step 1: Understand Matrix Subtraction

Matrix subtraction involves subtracting corresponding elements of two matrices of the same size. In this exercise, both matrices are 3x3, so subtraction is possible.

Step 2: Subtract Element-Wise

Subtract each corresponding element of the second matrix from the first:\[\begin{array}{rrr}-9 - (-1) & 2 - 3 & -7 - 6 \8 - (-7) & 10 - (-3) & 3 - 5 \-7 - 2 & 4 - 11 & 15 - (-4) \\end{array}\]

Step 3: Perform Calculations

Compute each element of the resulting matrix:- First row: - \(-9 - (-1) = -8\) - \(2 - 3 = -1\) - \(-7 - 6 = -13\)- Second row: - \(8 - (-7) = 15\) - \(10 - (-3) = 13\) - \(3 - 5 = -2\) - Third row: - \(-7 - 2 = -9\) - \(4 - 11 = -7\) - \(15 - (-4) = 19\)The resulting matrix is:\[\begin{array}{rrr}-8 & -1 & -13 \15 & 13 & -2 \-9 & -7 & 19 \\end{array}\]

Step 4: Confirm Solution

Verify each of the individual element calculations to ensure the matrix subtraction was performed correctly. If each step is consistent with matrix subtraction rules, then the solution is confirmed correct.

Key Concepts

Matrix Operations

Matrix operations encompass a set of mathematical procedures that include addition, subtraction, multiplication, and more, applied to arrays of numbers known as matrices. Each operation has specific rules and requirements that must be met to yield a valid result.
In the case of matrix subtraction, as in our exercise, it requires two matrices of identical dimensions. This means each matrix must contain the same number of rows and columns, allowing us to pair corresponding elements for the operation.

• Addition: Adding corresponding elements.
• Subtraction: Subtracting corresponding elements (focus of our exercise).
• Multiplication: More complex, involving sums of products.

This exercise uses subtraction, an operation that directly relates to combining matrices by correlating each element individually.

Element-wise Subtraction

Element-wise subtraction is a straightforward concept where you subtract elements from two matrices one by one. Each element in the first matrix is paired with the corresponding element in the second matrix.
Let's break it down into steps:

• Pair Elements: Identify corresponding elements in each matrix.
• Subtract: Calculate the difference for each paired element.
• Form Resulting Matrix: Compile the results into a new matrix.

For example, for two matrices with elements at the first position \(a_{11}\) and \(b_{11}\), the subtraction will be \(a_{11} - b_{11}\). This process is repeated for each element in the matrices.
In our specific example, pairs like \(-9\) and \(-1\) are subtracted to form the first element of the resulting matrix.

3x3 Matrices

A 3x3 matrix is a grid with three rows and three columns, comprising nine elements in total. This type of matrix is frequently encountered in various mathematical and real-world applications as it provides a compact yet manageable data structure.
Working with a matrix of this size keeps operations like matrix subtraction relatively simple and direct. Consider the following structure of a 3x3 matrix:
\[ \begin{array}{ccc} a_{11} & a_{12} & a_{13} \b_{11} & b_{12} & b_{13} \c_{11} & c_{12} & c_{13} \end{array} \]
This structure is replicated in both the matrices involved in our exercise, ensuring that each element has a corresponding counterpart in the opposite matrix for subtraction.

Algebra 2

Algebra 2 is a branch of mathematics dealing with complex equations, functions, and operations that form the foundation for advanced mathematical concepts. Matrix operations, such as subtraction, are an integral part of Algebra 2.
This level of algebra introduces students to new ways of manipulating numbers and expressions, providing practice with various algebraic techniques. The matrix operations you encountered form a building block for this complex subject.
By understanding element-wise subtraction of matrices, students can develop analytical skills crucial for more advanced topics. These skills are also applicable in fields like physics, computer science, and engineering.
Gaining familiarity with matrix operations prepares students for more intricate concepts and problem-solving approaches in higher-level math courses, proving its importance in the Algebra 2 curriculum.