Problem 53 - Algebra 2

Question

Perform the indicated operations. $$ \left[\begin{array}{rr}{3} & {3} \\ {0} & {-2}\end{array}\right]-\left[\begin{array}{rr}{2} & {-1} \\ {5} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

The result of the matrix subtraction is \( \left[ \begin{array}{cc} 1 & 4 \\ -5 & -4 \end{array} \right] \).

Step 1: Identify the Components

First, notice that we have two matrices from which we will subtract: \( \left[ \begin{array}{rr} 3 & 3 \ 0 & -2 \end{array} \right] \) and \( \left[ \begin{array}{rr} 2 & -1 \ 5 & 2 \end{array} \right] \). Each has dimensions of \( 2 \times 2 \).

Step 2: Perform Matrix Subtraction

Subtract corresponding elements of the matrices. For matrices \( A = \left[ \begin{array}{rr} 3 & 3 \ 0 & -2 \end{array} \right] \) and \( B = \left[ \begin{array}{rr} 2 & -1 \ 5 & 2 \end{array} \right] \), the subtraction \( A - B \) is performed element-wise: \[ \begin{array}{cc} 3-2 & 3-(-1) \ 0-5 & -2-2 \end{array} \] which simplifies to: \[ \begin{array}{cc} 1 & 4 \ -5 & -4 \end{array} \]

Step 3: Review the Result

Verify the resulting matrix: \[ \begin{array}{cc} 1 & 4 \ -5 & -4 \end{array} \]. Each element has been correctly calculated by subtracting elements of the first matrix from the corresponding elements of the second matrix.

Key Concepts

Matrix Operations

Matrix operations are mathematical procedures applied to matrices, including addition, subtraction, multiplication, and more. When performing operations like subtraction, each corresponding element of one matrix is directly subtracted from the other. This operation requires that both matrices have the same dimensions. Otherwise, they cannot be directly subtracted. In this exercise, subtraction of two matrices happens element-wise, meaning you subtract the element in the first row and first column of the first matrix from the element in the first row and first column of the second matrix, and so on.
It is important to carry out these operations correctly, ensuring that each element is paired with its counterpart in the other matrix. The process is straightforward and heavily relies on basic arithmetic operations. Subtraction can be directly visualized by stacking two matrices and then performing calculations accordingly for each pair.
Understanding matrix operations lays the foundation for more advanced concepts like solving systems of equations and transformations in algebra and calculus.

2x2 Matrices

A 2x2 matrix is a specific type of matrix that has two rows and two columns. These matrices are simple yet powerful, often forming the building blocks for understanding more complex matrix structures. In this exercise, both matrices used for subtraction are 2x2 matrices. This uniformity in size allows us to perform the subtraction operation as each element aligns perfectly with its counterpart.
In general, a 2x2 matrix can be written as: \[\begin{array}{cc}a & b \c & d\end{array}\] where \(a, b, c, d\) are the matrix elements.
Such matrices are commonly used to teach fundamental matrix operations and are useful in real-world applications such as rotations of objects in computer graphics and modeling simple economic systems. Simplifying operations involving 2x2 matrices helps students grasp important linear algebra concepts crucial for further studies.

Elementary Row Operations

Elementary row operations are transformations that can be performed on the rows of matrices. While not directly used in simple subtraction, understanding these operations is crucial as they form the backbone for more complex procedures such as Gaussian elimination. These transformations include:

• Swapping two rows,
• Multiplying a row by a non-zero scalar,
• Adding or subtracting the multiple of one row to another row.

Although our exercise focused on matrix subtraction, appreciation of these row operations helps improve problem-solving skills for larger systems of equations. Once students master operations on individual elements, they can progress to utilizing row operations to solve linear equations and find determinants. This more thorough understanding assists in both academic and real-world problem-solving scenarios, such as network analysis and optimization problems.