Question

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 matrices such that \(A C=B C\), then \(A\) is not necessarily equal to \(B\). Illustrate this using the following matrices. \(A=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 5 & 4 \\ 3 & -2 & 1\end{array}\right], B=\left[\begin{array}{rrr}4 & -6 & 3 \\ 5 & 4 & 4 \\ -1 & 0 & 1\end{array}\right]\), and \(C=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 4 & -2 & 3\end{array}\right]\)

Short answer

Upon multiplying with Matrix C, matrices A and B yield the same product but are not identical matrices. Thus, the second statement, that equal product of matrices doesn't necessarily imply identical factors, is validated.

Step-by-step solution

Check the Given Matrices

First thing is to check the given matrices. Matrix A = \(\left[\begin{array}{ccc}1 & 2 & 3 \ 0 & 5 & 4 \ 3 & -2 & 1\end{array}\right]\), Matrix B = \(\left[\begin{array}{ccc}4 & -6 & 3 \ 5 & 4 & 4 \ -1 & 0 & 1\end{array}\right]\) and Matrix C = \(\left[\begin{array}{ccc}0 & 0 & 0 \ 0 & 0 & 0 \ 4 & -2 & 3\end{array}\right]\)

Compute AC and BC

Matrix multiplication is performed in a row by column manner. For \(AC\), each element of the resultant matrix is obtained by multiplying corresponding elements of rows of A with columns of C followed by their addition. Similar operation is done for \(BC\).

Compare AC and BC

After calculating \(AC\) and \(BC\), if they turn out to be equal, the validity of the second statement is illustrated, i.e, even though \(AC = BC\), A and B are not identical matrices. Thus, with matrices multiplication, we cannot conclude that identical product implies identical factors.

Record Your Findings

It is important to note down what was discovered through the exercise. This can be formulated in a short statement or conclusion.

Key concepts

Matrices Inequality

Unlike the multiplication of real numbers, the product of two matrices does not always adhere to the same equality rules. Specifically, if we have real numbers such that a, b, and c where c is nonzero and a·c = b·c, we can deduce a = b. But for matrices, even if we have matrices A, B, and C and it holds that A·C = B·C, A is not necessarily equal to B.

This discrepancy arises from the fact that matrix multiplication is not commutative, and one cannot simply 'divide' by a matrix to isolate the other matrix. In the given exercise, C is a matrix with zero elements in its first two rows, which implies any matrix multiplied by C will result in a matrix with its first two rows as zero. Hence, A·C and B·C might produce the same matrix regardless of the differences between A and B.

We must also consider the concept of a zero matrix in this context. A zero matrix acts somewhat like the number zero in real number multiplication—any matrix multiplied by a zero matrix will result in a zero matrix, further muddying the waters of our initial assumption.

Matrix Operations

The exercise showcases an essential component of linear algebra: matrix operations. Matrix multiplication, one of the fundamental operations, combines two matrices to produce a third matrix. It is crucial to follow a particular order when performing the multiplication. The element in row i and column j of the resultant matrix is calculated by taking the dot product of the ith row of the first matrix and the jth column of the second matrix.

Calculating Matrix Products

To calculate AC and BC, you multiply each entry of the rows of A (or B) with the corresponding entries of the columns of C and add up those products. This operation is performed for every possible combination of rows from the first matrix and columns from the second matrix. It's important to highlight that unless the number of columns in the first matrix matches the number of rows in the second matrix, multiplication is not possible—the matrices are said to be 'incompatible' for the operation. The matrices in our exercise are compatible, so we can proceed to multiply them as demonstrated.

Associative Property of Matrices

The associative property is a well-known concept in arithmetic and algebra which states that when three or more numbers are multiplied, the grouping of the numbers does not affect the product. Interestingly, this property also holds true in matrix multiplication. It implies that for any three matrices A, B, and C, given they have the appropriate dimensions for multiplication, the equation (A·B)·C = A·(B·C) will always be true.

When applying the associative property, it's important to remember that matrix multiplication also requires the matrices to be compatible in size, as mentioned before. The associative property allows for simplification in calculations and flexibility in the computation process because it gives the freedom to group matrices in different ways without altering the product. However, despite the associative property, it's crucial to point out that matrix multiplication is not commutative —A·B is not necessarily equal to B·A. Hence, the order in multiplication still matters even though the grouping does not.