Problem 54 - College Algebra

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. $$ A=\left[\begin{array}{rrr}{-2} & {0} & {9} \\ {1} & {8} & {-3} \\ {0.5} & {4} & {5}\end{array}\right], B=\left[\begin{array}{ccc}{0.5} & {3} & {0} \\\ {-4} & {1} & {6} \\ {8} & {7} & {2}\end{array}\right], C=\left[\begin{array}{ccc}{1} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {1}\end{array}\right] $$ $$ A B C $$

Step-by-Step Solution

Answer

The product of matrices \( A \), \( B \), and \( C \) is \[ \left[\begin{array}{rrr}{117.5} & {51} & {117.5} \\ {84.5} & {26} & {84.5} \\ {54} & {35} & {54} \end{array}\right] \].

Step 1: Matrix Multiplication Compatibility

First, verify if the matrices are suitable for multiplication. Matrix multiplication is only possible when the number of columns in the first matrix equals the number of rows in the second matrix. Here, matrix \( A \) is a 3x3 matrix, \( B \) is also 3x3, and \( C \) is likewise 3x3, hence the multiplications \( AB \), \( BC \), and \( (AB)C \) are possible.

Step 2: Multiply Matrices A and B

Perform the multiplication \( AB \) first. The element in the i-th row and j-th column of the product matrix is obtained by multiplying the elements of the i-th row of \( A \) by the elements of the j-th column of \( B \) and summing these products. Calculate the elements row by row and column by column for the entire matrix:\[ AB = \left[\begin{array}{rrr}{65.5} & {51} & {12} \ {45.5} & {26} & {39} \ {23} & {35} & {31} \end{array}\right] \]

Step 3: Multiply Result with Matrix C

Take the result from Step 2 and multiply it by matrix \( C \). Using the same method as before, calculate each element of the final product matrix:\[ (AB)C = \left[\begin{array}{rrr}{117.5} & {51} & {117.5} \ {84.5} & {26} & {84.5} \ {54} & {35} & {54} \end{array}\right] \]

Step 4: Verify Solution Using a Calculator

Use a calculator to multiply \( A \), \( B \), and \( C \) to verify that each step was performed correctly and the resulting matrix matches the one obtained manually.

Key Concepts

Matrix Compatibility

Before attempting any matrix multiplication, it's important to determine if the matrices involved are compatible. Matrix compatibility depends on the dimensions of the matrices. For two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix. This rule helps ensure that each element in the resulting matrix is properly calculated from corresponding elements in the involved matrices.
Let's break this down:

• Matrix A has dimensions 3x3, which means it has 3 rows and 3 columns.
• Matrix B is also 3x3, indicating it has 3 rows and 3 columns.
• Matrix C is similarly a 3x3 matrix.

In the given exercise, we have to multiply matrices A, B, and C. For these matrices:

• The steps involve multiplying A by B, then the result with C.
• The multiplication AB is possible since A's column count equals B's row count (both are 3).
• The same compatibility condition applies for multiplying (AB) by C.

The matrices are indeed compatible, confirming that these operations are feasible.

Matrix Operations

Matrix operations, particularly multiplication, follow specific rules that aren't as straightforward as basic arithmetic operations on numbers. In matrix multiplication, we perform dot products of rows and columns to find the resulting matrix elements.
Here's how the multiplication proceeds:

• Each element in the result of multiplying two matrices is derived from the sum of products of elements from rows of the first matrix and columns of the second.
• To find an element in the ith row and jth column in the result, multiply each element of the ith row in the first matrix by the corresponding element of the jth column in the second matrix, then sum the products.
• It's important to align correctly the indices corresponding to the rows and columns to get the elements accurately.

For example, by applying this method:

• In multiplying A and B, the resulting matrix AB is calculated element by element, with the values given in the solution.
• The next step involves using the matrix AB obtained and applying the same approach to multiply by C, resulting in the final matrix (AB)C.

Efficiently performing matrix operations requires careful attention to detail in the multiplication process.

Matrix Algebra

Matrix algebra encompasses general operations on matrices, including addition, subtraction, and multiplication, alongside other properties like determinants and inverses that aren't in this exercise but are key to advanced applications. In the context of our problem, the main focus is on matrix multiplication.
Matrix algebra offers a powerful way to handle multiple linear equations, perform transformations, and manage data in various mathematical models. Understanding matrix algebra starts with getting familiar with matrices and how to manipulate them.
Some key points to remember in matrix algebra:

• Associative Property: For any matrices A, B, and C, the equation (AB)C = A(BC) holds when the operations are defined, showing order doesn't change the outcome.
• Matrix multiplication is not commutative, meaning AB generally does not equal BA.
• Identity matrix plays a role similar to the number 1 in multiplication, leaving a matrix unchanged when multiplied.

In this exercise, we've seen matrix algebra in action, particularly using the associative property to compute (AB)C. Understanding these foundational operations enhances problem-solving capabilities in various fields like physics, computer science, and engineering.