Problem 30 - Discrete Mathematics with Applications

Question

Let \(A, B,\) and \(C\) be any \(m \times n\) matrices, \(O\) the \(m \times n\) zero matrix, and \(c\) and \(d\) any real numbers. Prove each (see Theorem 3.12). $$(-1) A=-A$$

Step-by-Step Solution

Answer

To prove \((-1)A = -A\), we can use properties from Theorem 3.12 as follows: 1. Begin with property 5: \((c+d)A=cA+dA\), substitute \(c=-1\) and \(d=1\), giving us \((-1+1)A = (-1)A + A\). 2. Simplify the expression to \((0)A = (-1)A + A\). 3. Using property 6 of Theorem 3.12, substitute \(c=0\) and \(d=-1\) to derive \(0((-1)A) = (0)A\). 4. Since both equations have the same right-hand side \((0)A\), we can conclude that \((-1)A + A = 0((-1)A)\). Therefore, we can prove that \((-1)A = -A\).

Step 1: Define the problem

We are given a matrix A and we need to prove that \((-1)A = -A\). To do so, we will use the properties of scalar multiplication and matrix addition from Theorem 3.12.

Step 2: Apply property 1 of Theorem 3.12

Start by adding the \(m \times n\) zero matrix O to A and see if we can relate the result to our given problem. From property 1, \(A + O = A\).

Step 3: Apply property 5 of Theorem 3.12

Now apply property 5 to our given problem. We have to show that \((-1)A +A = O\). Using property 5, \[ (-1+1)A = (-1)A + A \]. Property 5 states that \[(c+d)A=cA+dA\], so we can substitute \(c=-1\) and \(d=1\).

Step 4: Simplify the expression

After applying property 5, our expression becomes: \[ (0)A = (-1)A + A\]. This means that multiplying matrix A by 0 would negate (-1)A.

Step 5: Apply property 6 of Theorem 3.12

From property 6 of Theorem 3.12, we have \(c(dA) = (cd)A\). We can substitute \(c=0\) and \(d=-1\) to arrive at the following: \[ 0((-1)A) = (0)A \]

Step 6: Prove the given equation

From Steps 4 and 5, we have established that \(0((-1)A) = (0)A\) and \((-1)A + A = (0)A\). Since both equations have the same right-hand side \((0)A\), hence we can conclude that \((-1)A + A = 0((-1)A)\). Considering the equation \((-1)A + A = 0((-1)A)\), we see that for every element of A, its negation exists. Hence, we conclude that \((-1)A = -A\).

Key Concepts

Matrix Addition

Matrix addition is one of the basic operations in matrix algebra, allowing us to combine two matrices into one. To perform matrix addition, the matrices must be of the same size, meaning they should have the same number of rows and columns. This is crucial because addition is done element-wise.
For two matrices, say, matrix A and matrix B:

• Each element of matrix A is added to the corresponding element of matrix B.
• The result is stored in a new matrix, often denoted as A + B.

For example, if \(A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}\) and \(B = \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix}\), then their addition will be:\[A + B = \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{pmatrix}\]It’s worth noting that matrix addition is both commutative, meaning \(A + B = B + A\), and associative, meaning \((A + B) + C = A + (B + C)\). These properties simplify computations, making matrix addition a flexible tool in algebra.

Scalar Multiplication

Scalar multiplication involves multiplying every entry of a matrix by a scalar, which is simply a real number. Suppose we have a matrix A and a scalar c:

• We multiply each element of matrix A by c, resulting in a new matrix.

This operation is crucial because it applies uniformly across all matrix elements, altering the matrix's magnitude but not its dimensions. For example, for a matrix \(A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}\) and a scalar \(c\), the scalar multiplication is:\[cA = \begin{pmatrix} c \cdot a_{11} & c \cdot a_{12} \ c \cdot a_{21} & c \cdot a_{22} \end{pmatrix}\]An interesting case arises when \(c = -1\). This flips the sign of every element in the matrix, yielding what is called the additive inverse. Scalar multiplication is also distributive over matrix addition, meaning \(c(A + B) = cA + cB\). This property is integral in simplifying and solving matrix equations.

Zero Matrix

A zero matrix, denoted as \(O\), is a special type of matrix where all entries are zero. It serves a crucial role similar to the number zero in arithmetic. A zero matrix can have any dimensions, such as a 2x2 or 3x3 matrix, but it keeps the same properties regardless of size. Key characteristics include:

• When added to any matrix of the same dimensions, it does not change that matrix. In other words, \(A + O = A\).
• Multiplying any matrix by the zero matrix results in a zero matrix.

These properties make the zero matrix the identity element for addition in matrix algebra. It also frequently appears in proving properties, such as in our earlier example where it helped demonstrate the identity \((-1)A + A = O\). Its simplicity and straightforward behavior make the zero matrix a foundational concept in matrix operations.

Theorem 3.12

Theorem 3.12 introduces essential properties of matrices related to scalar multiplication and addition. These properties are foundational rules that support complex proofs and equations in matrix algebra.

• Property 1: \(A + O = A\), where \(O\) is the zero matrix, which acts as the identity under addition.
• Property 5: \((c + d)A = cA + dA\). This states that a scalar sum multiplication can be distributed across a matrix.
• Property 6: \(c(dA) = (cd)A\). This property shows that scalars can be combined before or during multiplication with a matrix, giving the same result.

In our specific problem, these properties were crucial. They helped demonstrate that multiplying a matrix by \(-1\) results in the matrix's additive inverse, thus proving that \((-1)A = -A\). These properties shine in simplifying, analyzing, and proving various matrix operations, underlining their importance in linear algebra.