Problem 89 - Precalculus with Limits

Question

THINK ABOUT IT If \(a\), \(b\), and \(c\) are real numbers such that \(c \neq 0\) and \(ac = bc\), then \(a = b\). However, if \(A\), \(B\), and \(C\) are nonzero matrices such that \(AC=BC\), then \(A\) is not necessarily equal to \(B\). Illustrate this using the following matrices. \(A = \left[ \begin{array}{r} 0 & 1 \\ 0 & 1 \end{array} \right]\), \(B = \left[ \begin{array}{r} 1 & 0 \\ 1 & 0 \end{array} \right]\), \(C = \left[ \begin{array}{r} 2 & 3 \\ 2 & 3 \end{array} \right]\)

Step-by-Step Solution

Answer

The matrices A, B, and C provided in the question when multiplied correctly according to the rules of matrix multiplication, result in \(AC = BC\). However, the matrices A and B are clearly not equal, thereby illustrating the crucial aspect of matrix multiplication where \(AC = BC\) does not necessarily imply \(A = B\), which is unlike the property of multiplication for real numbers.

Step 1: Performing Matrix Multiplication AC

To compute the product AC, multiply each element of a row of first matrix A with each element of a column of the second matrix C and then add them up. The matrix A has two rows and matrix C has two columns, so we'll have two such computations. The computed product AC would be \(\begin{bmatrix} 4 & 6 \ 4 & 6 \end{bmatrix}\).

Step 2: Performing Matrix Multiplication BC

Repeat the same process as step 1 for matrices B and C to get matrix BC. After the calculations, matrix BC is found to be \(\begin{bmatrix} 4 & 6 \ 4 & 6 \end{bmatrix}\).

Step 3: Comparing the Resultant Matrices

Now, by comparing both matrices obtained from steps 1 and 2, we see that \(AC = BC\), but matrix A was not equal to matrix B, which is exactly the case we were supposed to illustrate.

Key Concepts

Matrix Equivalence

Matrix equivalence is a fascinating concept in linear algebra that might puzzle students initially, especially when compared to real numbers. When we say two matrices, say \(A\) and \(B\), are equivalent, it usually refers to some form of transformation or operation that makes them interchangeable under certain conditions. However, it's crucial to note that matrix equivalence does not imply that \(A = B\) in the same way as we would with numerical equations.

• For example, multiplying two matrices can produce a result where \(AC = BC\), yet it does not mean \(A = B\).
• Unlike multiplication of real numbers where division and multiplication are straightforward and reversible, matrices offer more complexity due to their multi-dimensional nature.

Understanding this distinction is key to solving problems involving matrix equivalence, and to fully grasp when matrices can be manipulated like real numbers and when they cannot.

Non-zero Matrices

Non-zero matrices are matrices in which at least one element is not zero. These play an important role in matrix multiplication, where the product of two matrices can lead to some counterintuitive results. Consider the case of non-zero matrices \(A, B,\) and \(C\) given in the exercise. Even though \(AC = BC\), it didn't result in \(A = B\).

• Matrix \(C\) didn't act as an 'invertible' matrix in this scenario, which is why it couldn't be used to deduce \(A = B\).
• This is primarily because not all non-zero matrices have an inverse—a crucial aspect when it comes to solving matrices as equations.

Non-zero matrices hence enlighten us on how matrix multiplication doesn't mimic the straightforward nature of arithmetic multiplication of specific real numbers, leading instead to richer, more complex systems.

Real Numbers in Algebra

In algebra, real numbers are used as a straightforward and predictable set, defined by properties such as closure under addition, subtraction, multiplication, and division (excluding division by zero). These characteristics make real numbers simpler to handle than matrices, especially in algebraic equations.

• An equation like \(ac = bc\) leading to \(a = b\) is an everyday affair in algebra.
• However, translating this property directly to matrices doesn't hold because matrices are not as 'invertible' as real numbers.

Understanding this difference helps students reconcile why behaviors and results they see with numbers might diverge when dealing with matrices, enhancing their comprehension of the broader scope of algebra.