Problem 63 - Algebra 1

Question

Find the sum or the difference of the matrices. $$ \left[\begin{array}{rr}4 & -8 \\\7 & 0\end{array}\right]-\left[\begin{array}{rr}-5 & 3 \\\\-5 & -7\end{array}\right] $$

Step-by-Step Solution

Answer

Hence, the subtraction of the two given matrices results in the matrix [9 -11 // 12 7]

Step 1: Identify the corresponding components of both matrices

For the first matrix [4 -8 // 7 0] and the second matrix [-5 3 // -5 -7], the corresponding components to subtract would be 4 from -5, -8 from 3, 7 from -5, and 0 from -7.

Step 2: Carry out the subtractions

Subtracting the corresponding components gives: (4 - -5), (-8 - 3), (7 - -5) and (0 - -7). That simplifies to: 4 + 5; -8 - 3; 7 + 5; and 0 + 7 respectively.

Step 3: Obtain the final matrix

Upon performing the above operations, we have [9 -11 // 12 7] as the resultant matrix.

Key Concepts

Matrix Subtraction

Matrix subtraction involves finding the difference between two matrices of the same dimensions. If two matrices are not of the same size, subtraction cannot be performed. In this concept, each element from the first matrix is subtracted by the corresponding element from the second matrix.

For example, consider we have two matrices, A and B. To subtract matrix B from matrix A, you need to ensure both matrices have the same order, say 2x2. Matrix subtraction is performed using the formula:

• If Matrix A = \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) and Matrix B = \( \begin{bmatrix} e & f \ g & h \end{bmatrix} \)
• Then, A - B = \( \begin{bmatrix} a-e & b-f \ c-g & d-h \end{bmatrix} \)

Matrix subtraction is one of the fundamental operations in linear algebra used in various applications, from solving systems of equations to transformations in physics.

Component-wise Subtraction

Component-wise subtraction is a simple yet crucial process in matrix subtraction. It involves subtracting corresponding components, or elements, of the matrices involved. Consider it as subtracting numbers in a similar position from each matrix, much like subtracting by columns in a spreadsheet.

To perform component-wise subtraction, follow these steps:

• Identify elements in the same row and column in both matrices.
• Subtract the element from the second matrix from the element in the first matrix.
• Continue subtracting component by component until all elements have been accounted for.

It's important to handle negative numbers and zero correctly during the operation. Remember, subtracting a negative number is the same as adding its positive counterpart.

Algebra in Matrix Operations

Matrix operations such as subtraction rely on foundational concepts of algebra. Understanding how numbers interact within matrices involves applying basic algebraic rules. When performing matrix subtraction, recognizing operations like adding and subtracting integers, especially with negative values, is essential.

Here’s how algebra plays into matrix operations:

• Subtraction involves careful attention to signs. If subtracting might involve negative numbers, algebra helps clarify that \( a - (-b) = a + b \).
• Balancing equations within matrices requires understanding basic algebraic properties, such as the distributive and associative properties.

Utilizing these algebraic concepts is vital in ensuring accurate results during complex matrix operations like solving matrix equations or performing transformations. Developing a strong algebra foundation offers a smoother pathway into mastering matrix arithmetic.