Problem 14 - Precalculus with Limits

Question

In Exercises 11-18, if possible, find (a) \(A+B\), (b) \(A-B\), (c ) \(3A\), and (d) \(3A-2B\). \(A = \left[ \begin{array}{r} 1 & -1 & 3 \\ 0 & 6 & 9 \end{array} \right]\), \(B = \left[ \begin{array}{r} -2 & 0 & -5 \\ -3 & 4 & -7 \end{array} \right]\)

Step-by-Step Solution

Answer

\[(a) A+B=\left[\begin{array}{ccc} -1 & -1 & -2 \ -3 & 10 & 2 \end{array}\right], (b) A-B=\left[\begin{array}{ccc} 3 & -1 & 8 \ 3 & 2 & 16 \end{array}\right], (c) 3A=\left[\begin{array}{ccc} 3 & -3 & 9 \ 0 & 18 & 27 \end{array}\right], (d) 3A-2B=\left[\begin{array}{ccc} 6 & -3 & 19 \ 6 & 80 & 175 \end{array}\right]\]

Step 1: Find A+B

To add two matrices, simply add corresponding entries together. Therefore, \(A + B\) would be \(\left[ \begin{array}{r}1+(-2) & -1+0 & 3+(-5) \ 0+(-3) & 6+4 & 9+(-7) \end{array} \right]\) which simply gives \(\left[\begin{array}{ccc} -1 & -1 & -2 \ -3 & 10 & 2 \end{array} \right]\)

Step 2: Find A-B

To subtract two matrices, simply subtract corresponding entries. Therefore, \(A-B\) would be \(\left[\begin{array}{ccc} 1-(-2) & -1-0 & 3-(-5) \ 0-(-3) & 6-4 & 9-(-7) \end{array}\right]\), which gives \(\left[\begin{array}{ccc} 3 & -1 & 8 \ 3 & 2 & 16 \end{array}\right]\)

Step 3: Find 3A

This asks for the scalar multiplication of A by 3. To find the scalar product of 3 and A, multiply each element in A by 3, so we obtain \(\left[\begin{array}{ccc} 3*1 & 3*-1 & 3*3 \ 3*0 & 3*6 & 3*9 \end{array}\right]\), which gives \(\left[\begin{array}{ccc} 3 & -3 & 9 \ 0 & 18 & 27 \end{array}\right]\)

Step 4: Find 3A-2B

This asks for the scalar product of A by 3 and B by 2, and then subtract B from A. This can be done by subtracting each corresponding element in the two matrices: \((3*\left[ \begin{array}{r} 1 & -1 & 3 \ 0 & 6 & 9 \end{array} \right]) - (2*\left[ \begin{array}{r} -2 & 0 & -5 \ -3 & 4 & -7 \end{array} \right]) = \left[ \begin{array}{r} 3*(-2) + 4 & -3*0 + 0 & 9*(-5) + 10 \ 0*(-3) + 6 & 18*4 + 8 & 27*(-7) + 14 \end{array} \right]\), which then yields \(\left[\begin{array}{ccc} 6 & -3 & 19 \ 6 & 80 & 175 \end{array}\right]\)

Key Concepts

Matrix Addition

Matrix addition is one of the fundamental operations in matrix algebra. It involves adding two matrices by adding their corresponding elements. For matrix addition to be possible, both matrices must be of the same dimensions. That means they should have the same number of rows and columns.

Here's how you perform matrix addition step-by-step:

• Align the two matrices so that their corresponding elements are in the same position.
• Add each pair of corresponding elements together.
• Place the sum of each pair in the corresponding position in the resulting matrix.

In our example, matrix A and matrix B are both 2x3 matrices, meaning they have 2 rows and 3 columns. To find A+B, we add the numbers in the same positions together. For instance, the element in the first row and first column of matrix A (which is 1) is added to the element in the first row and first column of matrix B (which is -2), giving us -1. We repeat this process for every pair of corresponding elements, resulting in a new matrix with the same dimensions.

Matrix Subtraction

Matrix subtraction is quite similar to matrix addition but, instead of adding, we subtract the corresponding elements of the matrices. Just like addition, subtraction also requires that the matrices have the same dimensions.

Here's how you perform matrix subtraction step-by-step:

• Line up the two matrices as you would for addition.
• Subtract each pair of corresponding elements.
• Place the difference of each pair in the corresponding position in the resulting matrix.

In our example, to calculate A-B, we subtract the elements of matrix B from the corresponding elements of matrix A. For instance, the top-left element of matrix A (1) minus the top-left element of matrix B (-2) gives us 3, and so on for each pair, leading to a new matrix.

Scalar Multiplication of Matrices

Scalar multiplication of matrices involves multiplying every element of a matrix by a scalar (a single number). This operation is useful when we need to scale a matrix up or down or combine it with other matrix operations, such as in the calculation of 3A-2B.

Here's how scalar multiplication works:

• Take a scalar, which can be any real number.
• Multiply each element of the matrix by this scalar.
• The resulting matrix maintains the same size as the original matrix, with each element being the product of the original element and the scalar.

For the exercise of calculating 3A, we multiply every element of matrix A by 3, producing a new matrix where each element is three times its original value.

Matrix Algebra

Matrix algebra is an extension of regular algebra into multiple dimensions. It encompasses operations like addition, subtraction, and scalar multiplication, which we have already discussed, along with other operations like multiplication of matrices, finding the determinant, and the inverse of a matrix.

Matrix algebra follows specific rules that ensure consistency and meaningful results. Some of these rules are similar to regular algebra, such as the commutative property for addition (A+B = B+A), but others are unique to matrix operations, such as the non-commutativity of multiplication (AB is not necessarily equal to BA).

Understanding the basics of matrix operations is crucial to more advanced studies in fields such as linear algebra, computer science, engineering, and more. These foundational skills enable us to solve systems of linear equations, transform geometric figures, and work with datasets in machine learning algorithms, among other applications.