Additive Inverse

In the world of mathematics, the concept of an additive inverse plays a crucial role in various fields, including algebra, complex numbers, and matrices. This blog aims to provide a detailed explanation of additive inverses, covering their properties, examples, and applications in different mathematical contexts.

1.0What is an Additive Inverse?

The additive inverse of a number a is another number that, when added to a, yields zero. Mathematically, if a is a real number, its additive inverse is denoted as –a, and the relationship can be expressed as:

a + (–a) = 0 

This concept extends beyond real numbers and applies to complex numbers, matrices, and other mathematical structures as well.

2.0Additive Inverse Property

The additive inverse property is an essential characteristic of mathematical systems, particularly in fields like algebra. This property states that every element in a set has a unique additive inverse within that set. This ensures that for any number x, the equation:

x + (–x) = 0 

is always satisfied. This property is fundamental in solving equations, simplifying expressions, and understanding algebraic structures.

3.0Additive Inverse of Complex Numbers

Complex numbers are represented as a + bi, where both a and b are real numbers, and i(iota) is the imaginary unit with the property i² = –1. For a complex number z = a + bi, its additive inverse is found by negating both the real part and the imaginary part. Therefore, the additive inverse of z is:

–z = –a – bi 

To verify, adding z and –z gives:

(a + bi) + (–a – bi) = 0 

This confirms that –z is indeed the additive inverse of z.

4.0Additive Inverse of Matrices

In linear algebra, the additive inverse of a matrix is crucial for solving equations and performing matrix operations. The additive inverse of a matrix A is another matrix –A such that when added together, they yield the zero matrix 0.

To determine the additive inverse of a matrix A, simply multiply each entry of the matrix by –1. If A is an m × n matrix with elements a_ij, the additive inverse –A is an m × n matrix with elements –a_ij. Mathematically,

A + (–A) = 0

where 0 is the m × n zero matrix with all elements equal to zero.

where –A is $-A=\left(\begin{array}{cccc}-a_{11}& -a_{12}& \cdots & -a_{1n}\\ -a_{21}& -a_{22}& \cdots & -a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ -a_{m1}& -a_{m2}& \cdots & -a_{mn}\end{array}\right)$

Example

Consider the matrix A: $A=\left(\begin{array}{ll}1& 2\\ 3& 4\end{array}\right)$

The additive inverse of A is: $-A=\left(\begin{array}{ll}-1& -2\\ -3& -4\end{array}\right)$

To verify, we add A and –A:

$A+(-A)=\left(\begin{array}{ll}1& 2\\ 3& 4\end{array}\right)+\left(\begin{array}{ll}-1& -2\\ -3& -4\end{array}\right)=\left(\begin{array}{ll}0& 0\\ 0& 0\end{array}\right)$

5.0Finding the Additive Inverse

To find the additive inverse of a number or object, simply negate the value. This means:

• For a real number x, the additive inverse is –x.
• For a complex number a + bi, the additive inverse is –a – bi.
• For a matrix, negate each element of the matrix.

6.0Additive Inverse Solved Examples

Let's look at some examples to illustrate the concept of additive inverses:

1. Real Numbers: 

• For 5, the additive inverse is –5.
• For –3.2, the additive inverse is 3.2.
• For x = 7, the additive inverse is –7.
• For x = –3.5, the additive inverse is 3.5.

2. Complex Numbers:

• For 3 + 4i, the additive inverse is –3 – 4i.
• For –2 – 5i, the additive inverse is 2 + 5i.
• For z = 4 + 5i, the additive inverse is –4 – 5i.
• For z = –2 – 3i, the additive inverse is 2 + 3i.

3. Matrices

• Let $A=\left(\begin{array}{ll}1& 2\\ 3& 4\end{array}\right)$, the additive inverse of A is $-A=\left(\begin{array}{ll}-1& -2\\ -3& -4\end{array}\right)$.
• Let $B=\left(\begin{array}{ccc}-2& 0& 3\\ 4& -1& 5\end{array}\right)$, the additive inverse of B is $-B=\left(\begin{array}{ccc}2& 0& -3\\ -4& 1& -5\end{array}\right)$.

7.0Additive Inverse Practice Questions

Real Numbers

1. Find the additive inverse of 12.
2. Find the additive inverse of –8.4.
3. If x = –15, what is the additive inverse of x?

Complex Numbers

1. Find the additive inverse of 3 + 7i.
2. Find the additive inverse of 5 – 9i.
3. If z = 4 – 2i, what is the additive inverse of z?

Matrices

1. Find the additive inverse of the matrix $\left(\begin{array}{cc}2& -1\\ 3& 4\end{array}\right)$.
2. Find the additive inverse of the matrix $\left(\begin{array}{ccc}0& 5& -3\\ 7& -2& 1\end{array}\right)$.
3. If $A=\left(\begin{array}{cc}-4& 6\\ 2& -8\end{array}\right)$what is the additive inverse of A?