Problem 35 - Precalculus: Functions and Graphs

Question

Let $$A=\left[\begin{array}{rrr}3 & -3 & 7 \\\2 & 6 & -2 \\\4 & 2 & 5\end{array}\right] \text { and }B=\left[\begin{array}{rrr}-9 & 5 & -8 \\\3 & -7 & 1 \\\\-1 & 2 & 6\end{array}\right]$$ Evaluate the matrix expression. $$A^{2}+B^{2}$$

Step-by-Step Solution

Answer

The result of \( A^2 + B^2 \) is a matrix: \[\begin{array}{rrr}135 & -109 & 91 \\-39 & 92 & -33 \\45 & 3 & 95\end{array}\right].

Step 1: Find A²

To find \( A^2 \), multiply matrix \( A \) by itself. The calculation is as follows: \[A \times A = \left[\begin{array}{rrr}3 & -3 & 7 \2 & 6 & -2 \4 & 2 & 5\end{array}\right] \times \left[\begin{array}{rrr}3 & -3 & 7 \2 & 6 & -2 \4 & 2 & 5\end{array}\right]\]Calculate each element of the resulting \( 3 \times 3 \) matrix:- 1st row, 1st column: \( 3 \times 3 + (-3) \times 2 + 7 \times 4 = 9 - 6 + 28 = 31 \)- 1st row, 2nd column: \( 3 \times (-3) + (-3) \times 6 + 7 \times 2 = -9 - 18 + 14 = -13 \)- 1st row, 3rd column: \( 3 \times 7 + (-3) \times (-2) + 7 \times 5 = 21 + 6 + 35 = 62 \)- 2nd row, 1st column: \( 2 \times 3 + 6 \times 2 + (-2) \times 4 = 6 + 12 - 8 = 10 \)- 2nd row, 2nd column: \( 2 \times -3 + 6 \times 6 + (-2) \times 2 = -6 + 36 - 4 = 26 \)- 2nd row, 3rd column: \( 2 \times 7 + 6 \times (-2) + (-2) \times 5 = 14 - 12 - 10 = -8 \)- 3rd row, 1st column: \( 4 \times 3 + 2 \times 2 + 5 \times 4 = 12 + 4 + 20 = 36 \)- 3rd row, 2nd column: \( 4 \times -3 + 2 \times 6 + 5 \times 2 = -12 + 12 + 10 = 10 \)- 3rd row, 3rd column: \( 4 \times 7 + 2 \times (-2) + 5 \times 5 = 28 - 4 + 25 = 49 \)So, \( A^2 = \left[\begin{array}{rrr}31 & -13 & 62 \10 & 26 & -8 \36 & 10 & 49\end{array}\right] \).

Step 2: Find B²

Similarly, multiply matrix \( B \) by itself:\[B \times B = \left[\begin{array}{rrr}-9 & 5 & -8 \3 & -7 & 1 \-1 & 2 & 6\end{array}\right] \times \left[\begin{array}{rrr}-9 & 5 & -8 \3 & -7 & 1 \-1 & 2 & 6\end{array}\right]\]Calculate each element of the resulting \( 3 \times 3 \) matrix:- 1st row, 1st column: \( (-9) \times (-9) + 5 \times 3 + (-8) \times (-1) = 81 + 15 + 8 = 104 \)- 1st row, 2nd column: \( (-9) \times 5 + 5 \times (-7) + (-8) \times 2 = -45 - 35 - 16 = -96 \)- 1st row, 3rd column: \( (-9) \times (-8) + 5 \times 1 + (-8) \times 6 = 72 + 5 - 48 = 29 \)- 2nd row, 1st column: \( 3 \times (-9) + (-7) \times 3 + 1 \times (-1) = -27 - 21 - 1 = -49 \)- 2nd row, 2nd column: \( 3 \times 5 + (-7) \times (-7) + 1 \times 2 = 15 + 49 + 2 = 66 \)- 2nd row, 3rd column: \( 3 \times (-8) + (-7) \times 1 + 1 \times 6 = -24 - 7 + 6 = -25 \)- 3rd row, 1st column: \( (-1) \times (-9) + 2 \times 3 + 6 \times (-1) = 9 + 6 - 6 = 9 \)- 3rd row, 2nd column: \( (-1) \times 5 + 2 \times (-7) + 6 \times 2 = -5 - 14 + 12 = -7 \)- 3rd row, 3rd column: \( (-1) \times (-8) + 2 \times 1 + 6 \times 6 = 8 + 2 + 36 = 46 \)So, \( B^2 = \left[\begin{array}{rrr}104 & -96 & 29 \-49 & 66 & -25 \9 & -7 & 46\end{array}\right] \).

Step 3: Add A² and B²

Add matrices \( A^2 \) and \( B^2 \):\[A^2 + B^2 = \left[\begin{array}{rrr}31 & -13 & 62 \10 & 26 & -8 \36 & 10 & 49\end{array}\right] + \left[\begin{array}{rrr}104 & -96 & 29 \-49 & 66 & -25 \9 & -7 & 46\end{array}\right]\]Calculate each element:- 1st row, 1st column: \( 31 + 104 = 135 \)- 1st row, 2nd column: \( -13 + (-96) = -109 \)- 1st row, 3rd column: \( 62 + 29 = 91 \)- 2nd row, 1st column: \( 10 + (-49) = -39 \)- 2nd row, 2nd column: \( 26 + 66 = 92 \)- 2nd row, 3rd column: \( -8 + (-25) = -33 \)- 3rd row, 1st column: \( 36 + 9 = 45 \)- 3rd row, 2nd column: \( 10 + (-7) = 3 \)- 3rd row, 3rd column: \( 49 + 46 = 95 \)Thus, \( A^2 + B^2 = \left[\begin{array}{rrr}135 & -109 & 91 \-39 & 92 & -33 \45 & 3 & 95\end{array}\right] \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It allows us to combine two matrices to produce a new matrix, with each element in the resultant matrix being a sum of products of corresponding elements. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. This ensures that the operation is valid.
For instance, when multiplying matrix \( A \) with itself which is denoted as \( A^2 \), we compute the sum of the products of the corresponding entries in the rows of the first matrix and the columns of the second:

• Each element in the new matrix is given by multiplying the elements of one row by the elements of one column and adding them up.
• For example, in computing \( A^2 \), the element in the first row and first column is calculated as \( 3 \times 3 + (-3) \times 2 + 7 \times 4 = 31 \).

This process is repeated for every element in the resulting matrix, which explains why matrix multiplication can be more complex compared to other arithmetic operations.

Matrix Addition

Matrix addition is a straightforward process and generally easier to grasp than multiplication. It involves adding two matrices of the same dimensions by adding the corresponding elements together.
In our example, after calculating \( A^2 \) and \( B^2 \), we add these two matrices. This is done element-wise:

• The element in the first row and first column of the resultant matrix is the sum of the first row, first column elements of \( A^2 \) and \( B^2 \), i.e., \( 31 + 104 = 135 \).
• The matrices must have the same number of rows and columns for this operation to be performed. If not, matrix addition is not possible.

The resulting matrix from the addition will have the same dimensions as the two original matrices.

Matrix Algebra

Matrix Algebra is the overarching field that encompasses operations like multiplication, addition, and numerous other manipulations with matrices. Understanding these operations allows us to solve complex systems of equations and perform various transformations.
Matrix algebra is crucial in fields like computer science, physics, and engineering. It helps in:

• Representing and solving linear systems of equations.
• Modeling transformations in graphics and robotics.
• Calculating network flows in operations research.

The operations we performed, \( A^2 + B^2 \), are standard procedures in matrix algebra. They illustrate how multiple operations may be combined to solve specific algebraic expressions involving matrices.