Problem 107 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About it If \(a, b,\) and \(c\) are real numbers such that \(c \neq 0\) and \(a c=b c,\) then \(a=b .\) However, if \(A, B\) and \(C\) are nonzero matrices such that \(A C=B C\), then \(A\) is not necessarily equal to \(B\). Illustrate this using the following matrices. $$A=\left[\begin{array}{ll} 0 & 1 \\ 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right], \quad C=\left[\begin{array}{ll} 2 & 3 \\ 2 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

Even if two matrices \(A, B\) satisfy \(A C=B C\) for a nonzero matrix \(C\), it doesn't necessarily mean that \(A=B\). This is illustrated here with the given matrices, in which \(AC=BC\), but \(A\) and \(B\) are certainly not equal.

Step 1: Matrix Multiplication

Begin by multiplying the matrices \(A\) and \(B\) with the matrix \(C\). For matrix multiplication, take each row of the first matrix and do an element by element multiplication with each column of the second matrix and then add them up.

Step 2: Calculate AC

\[A C=\left[\begin{array}{ll}0 & 1 \ 0 & 1\end{array}\right] \times \left[\begin{array}{ll}2 & 3 \ 2 & 3\end{array}\right]=\left[\begin{array}{ll}2 & 3 \ 2 & 3\end{array}\right]\] This is achieved by multiplying and adding elements row by row from the first matrix (A) with column by column from the second matrix (C). Thus, AC gives us a matrix [2 3; 2 3].

Step 3: Calculate BC

\[B C=\left[\begin{array}{ll}1 & 0 \ 1 & 0\end{array}\right] \times \left[\begin{array}{ll}2 & 3 \ 2 & 3\end{array}\right]=\left[\begin{array}{ll}2 & 3 \ 2 & 3\end{array}\right]\] Similarly, the multiplication and addition of elements row by row from the first matrix (B) with column by column from the second matrix (C) produces the matrix BC=[2 3; 2 3].

Step 4: Are A and B equal?

From the above steps, it's clear that \(AC=BC\); however, if you compare the matrices \(A\) and \(B\), you will see that they are not equal. In other words, even though \(AC=BC\), \(A\) is not equivalent to \(B\). This contradicts the cancellation law from number theory, which says that if \(ac=bc\) and \(c≠0\), then \(a=b\). In the case of matrices, this law does not hold.

Key Concepts

Matrices

Matrices are fundamental mathematical structures used to organize and manipulate data. They are typically rectangular arrays of numbers, with elements arranged in rows and columns. Every matrix has a specified size, given by its dimensions — number of rows by number of columns. For example, a 2x2 matrix has 2 rows and 2 columns:
\[\begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22}\end{bmatrix}\]
To work effectively with matrices, understanding how to perform operations like addition, subtraction, and multiplication is essential. Matrix multiplication is a key operation, where elements are multiplied and summed in a specific manner. It's crucial to note that matrix multiplication is not commutative; that is, \(A \cdot B eq B \cdot A\) in general.

• Matrix multiplication involves taking rows from the first matrix and columns from the second and summing the products.
• The result of multiplying a \(m\times n\) matrix by a \(n\times p\) matrix is an \(m\times p\) matrix.
• This operation underpins many applications in sciences and engineering fields.

Matrix Equality

Matrix equality is a simple but crucial concept in linear algebra. Two matrices, say \(A\) and \(B\), are equal if all their corresponding elements are equal. That means every element in row \(i\) and column \(j\) of matrix \(A\) must be identical to the element in row \(i\) and column \(j\) of matrix \(B\).
If:
\[\begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22}\end{bmatrix}= \begin{bmatrix}b_{11} & b_{12} \b_{21} & b_{22}\end{bmatrix}\]
Then, \(a_{11} = b_{11}, a_{12} = b_{12}, \ldots\)
Matrix equality plays a vital role when comparing results from different matrix operations. Even if two different operations on matrices lead to the same resultant matrix, it does not imply the matrices that were operated upon are equal themselves, unless each respective element matches.

Cancellation Law in Matrices

The Cancellation Law is a familiar concept from real numbers, stating that if \(ac = bc\) and \(c eq 0\), then \(a = b\). However, this law doesn’t universally apply in the realm of matrices. The law breaks down because, unlike real numbers, the zero matrix does not possess an inverse.
Consider matrices \(A\), \(B\), and \(C\) where \(AC = BC\) could have perfectly valid operations that yield the same matrix without \(A\) equaling \(B\). This is partly because matrix multiplication can lead to loss of information. For instance, matrix \(C\) might annihilate certain distinguishing features between \(A\) and \(B\), leaving identical products even though \(A\) and \(B\) are different.

• Matrix \(C\) doesn’t contribute to making \(AC = BC\) imply \(A = B\), mainly if \(C\) has dependent rows or columns (like having rows or columns being multiples of each other).
• The outcome illustrates how linear dependencies in rows/columns of matrices can mask differences.
• This emphasizes the need for caution when inferring properties from multiplied matrices within algebraic systems.