Problem 32 - Algebra and Trigonometry

Question

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{rrr} {1} & {-1} & {1} \\ {5} & {0} & {-2} \\ {3} & {-2} & {2} \end{array}\right], \quad B=\left[\begin{array}{rrr} {1} & {1} & {0} \\ {1} & {-4} & {5} \\ {3} & {-1} & {2} \end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix for AB is \[ \left[\begin{array}{ccc} {3} & {-2} & {7} \ {1} & {5} & {-4} \ {5} & {0} & {2} \end{array}\right] \]and for BA is \[ \left[\begin{array}{ccc} {6} & {-1} & {1} \ {23} & {7} & {-8} \ {19} & {-7} & {3} \end{array}\right] \]

Step 1: Compute AB

Since both A and B are 3x3 matrices, the multiplication AB is possible. Each entry in the resulting matrix is found by multiplying corresponding entries from the row of the first and column of the second, then adding them all. For example, the entry in the first row and first column of the resulting matrix will be: \( (1*1) + (-1*1) + (1*3) = 1 - 1 + 3 = 3 \). Performing the same calculation for each entry in the matrix gives the resulting matrix: \[ AB = \left[\begin{array}{ccc} {3} & {-2} & {7} \ {1} & {5} & {-4} \ {5} & {0} & {2} \end{array}\right] \]

Step 2: Compute BA

Now, let's compute BA. Similar to step 1, this involves multiplying corresponding entries from the row of the first and the column of the second, then adding them all. For example, the entry in the first row and first column of the resulting matrix will be: \( (1*1) + (1*5) + (0*3) = 1 + 5 = 6 \). Performing the same calculation for each entry in the matrix gives the resulting matrix: \[ BA = \left[\begin{array}{ccc} {6} & {-1} & {1} \ {23} & {7} & {-8} \ {19} & {-7} & {3} \end{array}\right] \]

Step 3: Review the results

After performing the two sets of multiplications, we have found the two resulting matrices, AB and BA. Despite the fact that the same matrices A and B were used in both multiplication operations, the resulting matrices are not equal, illustrating the fact that matrix multiplication is not commutative. In other words, changing the order in which matrices are multiplied can result in different resulting matrices.

Key Concepts

Matrices

A matrix is a rectangular arrangement of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called elements or entries. Matrices are foundational to various fields of mathematics, including linear algebra, and they're used extensively in science and engineering to represent and solve systems of linear equations, perform geometric transformations, and model various physical systems.

Matrices are denoted with uppercase letters like 'A' and 'B' and can have dimensions referred to by the number of rows and columns they contain. For instance, a 3x3 matrix like the ones provided in your exercise has three rows and three columns. When working with matrices, it's important to know that the operations on them, such as addition, subtraction, and multiplication, abide by specific rules that can differ from those of regular arithmetic.

Associative Property

The associative property is a fundamental property of mathematics that pertains to how operands are grouped in expressions. It states that within a calculation that only involves addition or multiplication, the order in which operations are performed does not change the final result. In the context of matrices, this property holds true for matrix multiplication when dealing with more than two matrices.

For example, if you have three matrices A, B, and C, the associative property ensures that \( (AB)C = A(BC) \). This is particularly useful when computing large products, as it gives flexibility in choosing the order of operations that might simplify the computation. However, while the grouping of matrices can be rearranged without affecting the product, this does not imply that the order of the matrices themselves can be switched, leading us to our next key concept.

Non-Commutative Operation

Matrix multiplication is a non-commutative operation, which means that the order in which you multiply matrices matters. In more straightforward terms, generally \( AB \) does not equal \( BA \). This non-commutative nature is starkly different from multiplication with real numbers where switching the order doesn't affect the result, for instance, \(2 \times 3 = 3 \times 2\).

The example exercise demonstrates the non-commutative property clearly: the matrices AB and BA yielded different results when their order was switched. Such behavior is important to understand because it can have significant consequences when applying matrix multiplication to solve actual problems in subjects such as physics, economics, and engineering. Whenever matrices are involved, it's crucial to maintain the correct sequence to ensure the accuracy of your results.