Problem 46 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) and \(B\) are matrices of the same size, then \(A-B=\) \(A+(-1) B\)

Step-by-Step Solution

Answer

The statement "If \(A\) and \(B\) are matrices of the same size, then \(A-B = A+(-1)B\)" is true. This is because subtracting matrix \(B\) from matrix \(A\) is equivalent to negating every element in matrix \(B\) and adding the resulting matrix to matrix \(A\), which is the same as adding a matrix formed by the scalar multiplication \((-1)B\) to matrix \(A\).

Step 1: Recall matrix addition and subtraction rules

We know that for two matrices to be added or subtracted, they must have the same dimensions. So, if \(A\) and \(B\) are the same size, then both \(A-B\) and \(A + (-1)B\) are defined.

Step 2: Recall matrix scalar multiplication rules

Matrix scalar multiplication involves multiplying every element in the matrix by a scalar value. In our statement, the scalar value is \(-1\).

Step 3: Compare \(A-B\) and \(A+(-1)B\)

Let's consider the result of \(A-B\). When we subtract matrix \(B\) from matrix \(A\), we are essentially negating every element in matrix \(B\), and then adding the negated values to the corresponding elements in matrix \(A\). Now, let's consider the result of \(A+(-1)B\). When we multiply matrix \(B\) by the scalar \(-1\), every element in matrix \(B\) is negated. Then, when we add the negated matrix \(B\) to matrix \(A\), we are adding the negated values to the corresponding elements in matrix \(A\). As we can see, in both cases, we are doing the same operation of adding negated elements of matrix \(B\) to the corresponding elements in matrix \(A\). This shows that: \[A-B = A+(-1)B\]

Step 4: Conclusion

The statement "If \(A\) and \(B\) are matrices of the same size, then \(A-B = A+(-1)B\)" is true. The reason it is true is because the subtraction of matrix \(B\) from matrix \(A\) is equivalent to negating every element in matrix \(B\) and adding the resulting matrix to matrix \(A\), which is the same process as adding a matrix formed by the scalar multiplication \((-1)B\) to matrix \(A\).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation that involves the element-wise addition of two matrices. For two matrices to be added, they must have the same dimensions. This means they must have the same number of rows and columns. Each element in the resulting matrix is found by adding the corresponding elements of the input matrices.

• If matrix A is of size \(m \times n\) and matrix B is also \(m \times n\), then the sum, \(A + B\), is a new matrix of size \(m \times n\).
• Element-wise addition implies that the element in the i-th row and j-th column of matrix A is added to the element in the i-th row and j-th column of matrix B.

Matrix addition is both commutative and associative. Commutativity means that \(A + B = B + A\), and associativity implies \((A + B) + C = A + (B + C)\).

Matrix Subtraction

Matrix subtraction, like matrix addition, requires that the matrices involved have the same dimensions. The subtraction of matrix B from matrix A, denoted as \(A - B\), is achieved by subtracting each element of matrix B from the corresponding element in matrix A.

• For example, if \(a_{ij}\) represents an element in matrix A and \(b_{ij}\) represents the corresponding element in matrix B, then \((A - B)_{ij} = a_{ij} - b_{ij}\).
• Matrix subtraction can also be thought of as adding the negative of the matrix, helping students understand that subtraction is inherently linked to addition.

A handy tip is to view subtraction as the addition of a negative: \(A - B = A + (-B)\). This perspective simplifies understanding and operations.

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar, which is a constant number. This operation is straightforward but pivotal as it scales the entire matrix by the same factor.

• If A is a matrix and k is a scalar, then \(kA\) means we multiply every element \(a_{ij}\) in A by k.
• This operation is very useful for manipulating matrices and can transform a matrix by increasing or decreasing its values proportionally.

In the given exercise, multiplying matrix B by \(-1\) results in changing the sign of every element in matrix B. This is what links scalar multiplication closely with subtraction, making \((-1)B\) equivalent to subtracting B from another matrix.

Matrix Dimensions

Understanding matrix dimensions is crucial for anyone dealing with matrices. The dimension of a matrix is expressed as \(m \times n\), where m is the number of rows and n is the number of columns.

• A matrix can only be added to or subtracted from another matrix if they share the same dimensions. This condition ensures every element has a counterpart in the second matrix to pair with.
• In matrix multiplication scenarios, different rules apply, often involving the number of columns in the first matrix equaling the number of rows in the second matrix.

Remember, understanding dimensions accurately helps avoid mistakes during operations and is essential for legal matrix manipulations.