Problem 88 - Precalculus with Limits

Question

Use the following matrices to find \(AB\), \(BA\), \((AB)C\), and \(A(BC)\). What do your results tell you about matrix multiplication, commutativity, and associativity? \(A = \left[ \begin{array}{r} 1 & 2 \\ 3 & 4 \end{array} \right]\), \(B = \left[ \begin{array}{r} 0 & 1 \\ 2 & 3 \end{array} \right]\), \(C = \left[ \begin{array}{r} 3 & 0 \\ 0 & 1 \end{array} \right]\)

Step-by-Step Solution

Answer

The result of the matrix multiplication \(AB\) is \(\left[ \begin{array}{cc} 4 & 7 \ 8 & 15 \end{array} \right]\) while the result of \(BA\) is \(\left[ \begin{array}{cc} 3 & 4 \ 11 & 16 \end{array} \right]\). This shows that matrix multiplication is not commutative, meaning it matters the order you multiply these matrices. However, the result of \((AB)C\) is \(\left[ \begin{array}{cc} 12 & 7 \ 24 & 15 \end{array} \right]\), and the result of \(A(BC)\) is also \(\left[ \begin{array}{cc} 12 & 7 \ 24 & 15 \end{array} \right]\), which demonstrates that matrix multiplication is associative, meaning it doesn't matter if we perform the multiplication on \(A\) and \(B\) first or \(B\) and \(C\) first.

Step 1: Compute \(AB\)

Matrix \(AB\) is computed as follows: \[ AB = A \cdot B = \left[ \begin{array}{cc} 1\cdot0+2\cdot2 & 1\cdot1+2\cdot3 \ 3\cdot0+4\cdot2 & 3\cdot1+4\cdot3 \end{array} \right] = \left[ \begin{array}{cc} 4 & 7 \ 8 & 15 \end{array} \right] \]

Step 2: Compute \(BA\)

Matrix \(BA\) is computed as follows: \[ BA = B \cdot A = \left[ \begin{array}{cc} 0\cdot1+1\cdot3 & 0\cdot2+1\cdot4 \ 2\cdot1+3\cdot3 & 2\cdot2+3\cdot4 \end{array} \right] = \left[ \begin{array}{cc} 3 & 4 \ 11 & 16 \end{array} \right] \]. It can be observed that \(AB\) ≠ \(BA\), indicating that matrix multiplication is not commutative.

Step 3: Compute \((AB)C\)

First, note that \(AB\) was calculated in Step 1 to be: \(\left[ \begin{array}{cc} 4 & 7 \ 8 & 15 \end{array} \right]\). Thus, \((AB)C\) is computed as follows: \[ (AB)C = (AB) \cdot C = \left[ \begin{array}{cc} 4\cdot3+7\cdot0 & 4\cdot0+7\cdot1 \ 8\cdot3+15\cdot0 & 8\cdot0+15\cdot1 \end{array} \right] = \left[ \begin{array}{cc} 12 & 7 \ 24 & 15 \end{array} \right] \]

Step 4: Compute \(A(BC)\)

First, compute the product \(BC\) as follows: \[ BC = B \cdot C = \left[ \begin{array}{cc} 0\cdot3+1\cdot0 & 0\cdot0+1\cdot1 \ 2\cdot3+3\cdot0 & 2\cdot0+3\cdot1 \end{array} \right] = \left[ \begin{array}{cc} 0 & 1 \ 6 & 3 \end{array} \right] \]. Then, compute \(A(BC)\) as follows: \[ A(BC) = A \cdot (BC) = \left[ \begin{array}{cc} 1\cdot0+2\cdot6 & 1\cdot1+2\cdot3 \ 3\cdot0+4\cdot6 & 3\cdot1+4\cdot3 \end{array} \right] = \left[ \begin{array}{cc} 12 & 7 \ 24 & 15 \end{array} \right] \]. It can be observed that \((AB)C\) = \(A(BC)\), indicating that matrix multiplication is associative.

Key Concepts

Commutativity

Commutativity is a fundamental concept in mathematics that typically applies to addition and multiplication. In these operations, commutativity means that changing the order of the operands does not change the outcome. For example, in arithmetic, we know that

• 2 + 3 = 3 + 2
• 4 \times 5 = 5 \times 4

However, when it comes to matrix multiplication, this property does not generally hold. This is clearly demonstrated in the exercise where we encountered matrix multiplication with matrices A and B. We found that:

• \( AB = \begin{bmatrix} 4 & 7 \ 8 & 15 \end{bmatrix} \)
• \( BA = \begin{bmatrix} 3 & 4 \ 11 & 16 \end{bmatrix} \)

Clearly, \(AB eq BA\), illustrating that the order of multiplying matrices matters and matrix multiplication is not commutative. This is an important concept to remember when dealing with matrices, as unlike numbers, swapping them may yield a completely different matrix altogether.

Associativity

While matrix multiplication is not commutative, it does retain the property of associativity. Associativity refers to the manner in which operations are grouped. In simpler terms, if three numbers or expressions are being multiplied, the outcome remains the same even if their grouping changes. Mathematically, this is expressed as:

• If \( a, b, \) and \( c \) are numbers, then \( (a \times b) \times c = a \times (b \times c) \)

When we look at matrix multiplication, associativity remains valid. In the given example, we calculated \((AB)C\) and \(A(BC)\) and found that

• \((AB)C = \begin{bmatrix} 12 & 7 \ 24 & 15 \end{bmatrix}\)
• \(A(BC) = \begin{bmatrix} 12 & 7 \ 24 & 15 \end{bmatrix}\)

As both expressions yield the same result, this confirms the associativity of matrix multiplication. This means that when dealing with several matrices, you can choose which pair to multiply first without changing the final product.

Matrices

Matrices are an essential part of higher mathematics, especially useful in many branches such as algebra, calculus, and more. Essentially, a matrix is a rectangular array of numbers arranged in rows and columns. When working with matrices, there are specific operations you can perform, and one of the most important is matrix multiplication. Unlike number multiplication, matrix multiplication has a set of strict rules and usually results in a matrix different from ordinary number multiplication.
For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second. This is crucial as each element in the product matrix is the sum product of the corresponding row of the first matrix and the column of the second. Moreover, matrix multiplication can be used to solve linear equations, transform geometrical shapes, and in computer graphics and machine learning.

• Matrices are defined by their size: rows \( \times\) columns.
• To multiply matrices \( A \) and \( B \) such that \( AB \) is defined, the number of columns in \( A \) must be the same as the number of rows in \( B \).
• Matrix multiplication is associative but not commutative.

Understanding matrices and matrix multiplication is fundamental for anyone delving into advanced studies and applications in mathematics and science.