Question

The subtraction of a matrix \(B\) may be considered as the addition of the matrix (-1)\(B\). Does the commutative law of addition permit us to state that \(A-B=B-A ?\) If not, how would you correct the statement?

Short answer

No, typically \(A - B \neq B - A\), except when \(A = B\).

Step-by-step solution

Understanding Matrix Subtraction

Matrix subtraction, such as \(A - B\), can be expressed as the addition of matrices \(A + (-B)\), where \(-B\) is the matrix with each element of \(B\) multiplied by \(-1\). This sets the stage for understanding the problem in terms of addition.

Commutative Property of Matrix Addition

The commutative property states that for any matrices \(X\) and \(Y\), \(X + Y = Y + X\). This property is true for matrix addition, meaning the order of addition does not affect the result.

Checking the Given Expression

The expression \(A - B = B - A\) translates to \(A + (-B) = B + (-A)\) using the idea that subtraction is equivalent to adding the negative of a matrix. Here, the expectation is that commutative property can apply. But if evaluated, \(A + (-B)\) typically will not equal \(B + (-A)\).

Conclusion from Evaluation

Since \(A + (-B)\) and \(B + (-A)\) are generally not equal (except when \(A = B\)), the expression does not normally satisfy the commutative property. Thus, \(A - B eq B - A\).

Correcting the Statement

To correct the statement, we should say that \(A + (-B) eq B + (-A)\) unless \(A = B\), at which point both sides would equal a zero matrix.

Key concepts

Matrix Subtraction

Matrix subtraction is an operation where one matrix is subtracted from another. Unlike regular arithmetic, subtraction in matrices is simply turning it into an addition problem. In essence, subtracting matrix \( B \) from matrix \( A \) is the same as adding the negative of \( B \) to \( A \). Instead of dealing with subtraction directly, you treat it as adding matrix \( A \) with matrix \(-B\), where \(-B\) is constructed by multiplying every element of \( B \) by -1.
This operation is very useful because it turns the task into one that relies on the addition of matrices, which follows well-established properties that are easier to work with. By converting subtraction into addition, you can easily employ concepts like the negative matrix to simplify and solve problems that involve removing elements of another matrix.

Commutative Property

The commutative property is a fundamental principle in mathematics. It tells us that when adding numbers or matrices, the order does not affect the sum. For matrices, this means if you have two matrices \( X \) and \( Y \), then \( X + Y = Y + X \). This is a handy feature that applies specifically to addition, simplifying many calculations.
However, when you translate subtraction into matrix addition by introducing a negative matrix, as in \( A - B = A + (-B) \), the commutative property does not apply to the original subtraction operation. Thus, the statement \( A - B = B - A \) fails to hold generally.
Understanding these differences is crucial, as mistaken application of this property in subtraction scenarios can lead to wrong conclusions in mathematical work and analysis.

Matrix Addition

Matrix addition involves combining two matrices of the same dimensions by adding their corresponding elements. So, if you have matrices \( A \) and \( B \), every element in the resulting matrix \( C = A + B \) is formed by adding elements \( a_{ij} + b_{ij} \), where \( a_{ij} \) and \( b_{ij} \) are elements from the respective matrices at position \( i \) and \( j \).
This operation is straightforward, as long as attention is paid to ensure that the matrices share the same dimensions. If the matrices have different dimensions, this operation cannot be performed.
Matrix addition is one of the core matrix operations and serves as a building block for more complex operations, like matrix subtraction when viewed as \( A + (-B) \). It is vital to be comfortable with addition as it plays an intermediary role in various mathematical manipulations.

Negative Matrix

A negative matrix, denoted as \(-B\), is created by taking each element of matrix \( B \) and multiplying it by -1. For instance, if the matrix \( B \) is:

• \( b_{11} = 3 \)
• \( b_{12} = -2 \)

Then the negative matrix \(-B\) would be:

• \(-b_{11} = -3 \)
• \(-b_{12} = 2 \)

This concept is crucial in transforming subtraction into addition in matrix operations.
Using a negative matrix allows one to use addition's properties even when doing an operation that originated as subtraction. By converting \( A - B \) to \( A + (-B) \), you can more easily apply properties like the associative and distributive laws in algebraic manipulations. This simplification is a powerful tool and essential for working with more complex algebraic expressions in matrices.