Problem 6 - Algebra 1

Question

Find the sum and the difference of the matrices. $$ \left[\begin{array}{rrr} 1 & 8 & -2 \\ -4 & -5 & 6 \end{array}\right],\left[\begin{array}{rrr} -1 & 9 & 2 \\ 3 & 3 & -5 \end{array}\right] $$

Step-by-Step Solution

Answer

The sum of the two matrices is \[ \left[\begin{array}{rrr} 0 & 17 & 0 \ -1 & -2 & 1 \end{array}\right] \] and the difference is \[ \left[\begin{array}{rrr} 2 & -1 & -4 \ -7 & -8 & 11 \end{array}\right] \]

Step 1: Find the sum of the matrices

Add corresponding elements together. \[ \left[\begin{array}{rrr} 1 & 8 & -2 \ -4 & -5 & 6 \end{array}\right] + \left[\begin{array}{rrr} -1 & 9 & 2 \ 3 & 3 & -5 \end{array}\right] = \left[\begin{array}{rrr} 1-1 & 8+9 & -2+2 \ -4+3 & -5+3 & 6-5 \end{array}\right] \]

Step 2: Calculate the values

Perform the operations for each element to get the resultant matrix.\[ \left[\begin{array}{rrr} 1-1 & 8+9 & -2+2 \ -4+3 & -5+3 & 6-5 \end{array}\right] = \left[\begin{array}{rrr} 0 & 17 & 0 \ -1 & -2 & 1 \end{array}\right] \]

Step 3: Find the difference of the matrices

Subtract corresponding elements. \[ \left[\begin{array}{rrr} 1 & 8 & -2 \ -4 & -5 & 6 \end{array}\right] - \left[\begin{array}{rrr} -1 & 9 & 2 \ 3 & 3 & -5 \end{array}\right] = \left[\begin{array}{rrr} 1-(-1) & 8-9 & -2-2 \ -4-3 & -5-3 & 6-(-5) \end{array}\right] \]

Step 4: Calculate the values

Perform the operations for each element to get the resultant matrix.\[ \left[\begin{array}{rrr} 1-(-1) & 8-9 & -2-2 \ -4-3 & -5-3 & 6-(-5) \end{array}\right] = \left[\begin{array}{rrr} 2 & -1 & -4 \ -7 & -8 & 11 \end{array}\right] \]

Key Concepts

Operations with Matrices

Matrix operations are procedures that can be performed on matrices, including addition, subtraction, and multiplication. These are fundamental tools used in various fields such as physics, computer science, and economics to model and solve complex problems.

It's crucial to understand that these operations follow specific rules. For instance, matrices need to be of the same dimensions to be added or subtracted. To make this concept easier to grasp, think of matrices like blocks of numbers that can only be combined with other blocks of the same size, much like pieces of a puzzle fitting together. Understanding these rules is essential for performing matrix operations successfully.

Matrix Arithmetic

Matrix arithmetic involves performing basic arithmetic operations, such as addition, subtraction, and multiplication, on matrices. Unlike addition and subtraction, which are performed element-wise, multiplication follows a special set of rules.

In matrix arithmetic, it's important to remember that the position of elements affects the outcome of operations. When adding or subtracting matrices, you do this element by element, aligning the entries based on their positions. But in multiplication, the position of elements in one matrix corresponds to rows and columns in another. Think of matrix arithmetic as a system that extends traditional arithmetic into a form that can handle data in arrays.

Element-wise Addition

Understanding Element-wise Addition

Element-wise addition refers to the process of adding two matrices by combining corresponding elements. To successfully perform this operation, it's imperative that both matrices have the same dimensions. Imagine a checkerboard where each square is a matrix element; to add them, you'd simply combine squares that match in position.

For example, if we have two matrices A and B where A contains a number at a particular position and B contains another number at the exact same position, we would add the numbers together to get the sum matrix C at that position. It's a straightforward process, but attention to detail is critical to ensure accuracy.

Element-wise Subtraction

Understanding Element-wise Subtraction

Element-wise subtraction is similar to addition but involves taking the difference between corresponding elements of matrices. Again, this operation requires that both matrices have identical dimensions. Think of this as having two stacks of colored blocks, each color corresponding to a different matrix; to find the difference, you would subtract the number of blocks in one stack from the other, color by color, position by position.

When we subtract one matrix from another, we're essentially removing the value of an element in the second matrix from the corresponding element in the first matrix. This is used in various applications, such as image processing, to modify or filter out details.