Problem 21 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The resulting matrix is \( \left[ \begin{array}{rr} 5 & 5 \\ 12 & 0 \end{array} \right] \).

Step 1: Understanding Matrix Addition

Matrix addition involves adding corresponding elements from each matrix. Ensure that both matrices are of the same dimensions. These two matrices are both 2x2 matrices, so we can add them.

Step 2: Adding Corresponding Elements

Take each element from the first matrix and add it to the element in the same position from the second matrix. \[ \begin{align*} (6 + (-1)), & (-2 + 7) \ (5 + 7), & (4 + (-4)) \end{align*} \]

Step 3: Calculating the Sum of Elements

Perform the arithmetic operations calculated in the previous step. \[ \begin{align*} 6 + (-1) & = 5 \ -2 + 7 & = 5 \ 5 + 7 & = 12 \ 4 + (-4) & = 0 \end{align*} \]

Step 4: Writing the Resulting Matrix

Use the sums from the calculations to form the resulting 2x2 matrix. \[ \left[ \begin{array}{rr} 5 & 5 \ 12 & 0 \end{array} \right] \]

Key Concepts

Matrix Operations

Matrix operations are essential tools in linear algebra that allow us to modify and transform arrays of numbers in meaningful ways. These operations can include tasks such as addition, subtraction, and multiplication of matrices. An important rule when performing any operation on matrices is that they must be of compatible sizes. For instance, matrix addition is only possible if both matrices have the same dimensions.

When adding two matrices together, we simply add the corresponding elements from each matrix. This means if you have two matrices, each with entries in the same position, you pair those numbers together for the operation. The result is a new matrix with the same dimensions as the original ones. For example, if you have two matrices that are both 2x2, after adding them, you'll still have a 2x2 matrix.

Understanding matrix operations gives you a foundation for more complex operations like matrix multiplication, inversion, and other transformations. These are widely used in various applications like computer graphics, physics, and statistics.

2x2 Matrices

The 2x2 matrix is one of the simplest forms of a matrix in linear algebra. It contains two rows and two columns which makes it easy to manipulate, especially when performing matrix arithmetic operations. A 2x2 matrix is conventionally written with square brackets, for example: \[ \begin{array}{cc} a & b \ c & d \end{array} \]This matrix format holds four elements, named as \(a\), \(b\), \(c\), and \(d\), each occupying a unique position.

The simplicity of 2x2 matrices makes them ideal for educational purposes, offering a gentle introduction to concepts like determinant and inverse calculations. Their small size simplifies the arithmetic operations as well, making them perfect for practice.

In our given example problem, we have matrices that are both 2x2. This makes it possible to perform matrix addition, an operation where we simply add the numbers at corresponding positions together. This results in another 2x2 matrix, maintaining the matrix's original dimensions.

Matrix Arithmetic

Matrix arithmetic involves basic mathematical operations such as addition, subtraction, and multiplication but executed under specific rules that pertain to matrices. These operations are foundational for computational tasks in both theoretical and applied mathematics.

When performing matrix arithmetic, especially addition or subtraction, you must always keep element positions aligned. This means each element in one matrix is added or subtracted from the element in the exact same position in the other matrix. Failing to align these positions correctly will lead to errors.

The arithmetic rules hold constant regardless of the values; it's the structure and alignment that remain pivotal. For example, in our matrix addition problem, to get the final matrices, each element in the resultant matrix corresponds to the direct sum of elements in the corresponding position of the two initial matrices. Matrix arithmetic proves to be a powerful tool in scenarios ranging from solving systems of equations to transformations in vectors.