Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Operations with Matrices Find, if possible, \((a) A+B,(b) A-B,(c) 3 A,\) and \((d) 3 A-2 B.\) Use the matrix capabilities of a graphing utility to verify your results. $$A=\left[\begin{array}{rr} 5 & -2 \\ 3 & 1 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & 1 \\ -2 & 6 \end{array}\right]$$

Step-by-Step Solution

Answer

The calculated matrices are (a) \( A + B = \left[\begin{array}{rr} 8 & -1 \\ 1 & 7 \end{array}\right] \), (b) \( A - B = \left[\begin{array}{rr} 2 & -3 \\ 5 & -5 \end{array}\right] \), (c) \( 3A = \left[\begin{array}{rr} 15 & -6 \\ 9 & 3 \end{array}\right] \), and (d) \( 3A - 2B = \left[\begin{array}{rr} 9 & -8 \\ 13 & -9 \end{array}\right] \)

Step 1: Calculate A + B

To add matrices, simply add corresponding entries together: \( A + B = \left[\begin{array}{rr} 5+3 & -2+1 \\ 3-2 & 1+6 \end{array}\right] = \left[\begin{array}{rr} 8 & -1 \\ 1 & 7 \end{array}\right] \)

Step 2: Calculate A - B

Similarly, to subtract matrices, subtract corresponding entries: \( A - B = \left[\begin{array}{rr} 5-3 & -2-1 \\ 3+2 & 1-6 \end{array}\right] = \left[\begin{array}{rr} 2 & -3 \\ 5 & -5 \end{array}\right] \)

Step 3: Perform Scalar Multiplication 3A

To calculate 3A, multiply every entry in A by 3: \( 3A = 3 \times \left[\begin{array}{rr} 5 & -2 \\ 3 & 1 \end{array}\right] = \left[\begin{array}{rr} 15 & -6 \\ 9 & 3 \end{array}\right] \)

Step 4: Perform Matrix Operation 3A - 2B

First perform the scalar multiplications 3A and 2B, and then subtract the results: \( 3A - 2B = (3 \times A) - (2 \times B) = \left[\begin{array}{rr} 15 - 6 & -6 - 2 \\ 9 + 4 & 3 - 12 \end{array}\right] = \left[\begin{array}{rr} 9 & -8 \\ 13 & -9 \end{array}\right] \)

Key Concepts

Matrix Arithmetic

Matrix arithmetic is a fundamental concept in linear algebra involving operations that can be performed on matrices, which are rectangular arrays of numbers. Common operations include addition, subtraction, and multiplication (both scalar and matrix multiplication).

When dealing with matrix arithmetic, it's essential to remember that operations are performed element-wise except for matrix multiplication. This means that for addition and subtraction, matrices must have the same dimensions, whereas for matrix multiplication, specific size conditions must be met. The ease of these operations on paper translates well into computational tasks, making them a basic but powerful tool in various applications ranging from solving systems of equations to computer graphics and data analysis.

Scalar Multiplication of Matrices

Scalar multiplication of matrices involves multiplying every element of a matrix by a constant value called a scalar. For example, if you have a scalar denoted as 'c' and a matrix 'M', when you perform scalar multiplication, you'll compute a new matrix where each entry 'm_{ij}' of 'M' is multiplied by 'c'.

Mathematically, it's represented as 'cM'. The operation is straightforward and doesn't change the size or shape of the original matrix but merely scales its entries. Scalar multiplication is often used in scaling geometric objects, adjusting data sets, and as a component of more complex matrix operations.

Matrix Addition and Subtraction

Matrix addition and subtraction are similar to adding and subtracting numbers, but they're performed element by element. To add or subtract two matrices, they must be of the same size; that is, they must have the same number of rows and columns. In matrix addition, corresponding elements from each matrix are added to form a new matrix. Conversely, in matrix subtraction, the corresponding elements are subtracted.

This operation is vital in many areas, including the solution of linear systems where one might need to add or subtract equations represented in matrix form. The simplicity of these operations also extends to statistical analysis and theoretical mathematics, where matrices are used to represent complex systems.

Using Graphing Calculators for Matrices

Graphing calculators are incredibly useful tools for performing matrix operations, especially for larger matrices or complex computations that would be tedious to do by hand. Most graphing calculators have dedicated functions to handle matrix arithmetic, allowing you to input matrices and perform operations like addition, subtraction, scalar multiplication, and even more advanced processes with relative ease.

They are a boon for students and professionals alike as they save time and reduce the potential for manual calculation errors. Additionally, graphing calculators can be employed to visualize data transformations that result from matrix operations, helping to connect abstract concepts with concrete visual representations. Always double-check that the matrices are input correctly to avoid any errors in the calculations.