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Additive Inverse of a Matrix — Definition & Examples

Additive Inverse of a Matrix

The matrix obtained by changing the sign of every matrix element. The additive inverse of matrix A is written –A.

Note: The sum of a matrix and its additive inverse is the zero matrix.

\[A = \left[ {\begin{array}{*{20}{c}}2&{ - 5}&4\\1&0&{ - 11}\end{array}} \right]\] \[ - A = \left[ {\begin{array}{*{20}{c}}{ - 2}&5&{ - 4}\\{ - 1}&0&{11}\end{array}} \right]\]            

See also

Matrix addition

Key Formula

A+(−A)=OA + (-A) = O
Where:
  • AA = Any matrix of size m × n
  • −A-A = The additive inverse of A, formed by negating every element of A
  • OO = The zero matrix of the same size m × n (all elements are 0)

Worked Example

Problem: Find the additive inverse of the matrix A and verify that A + (−A) equals the zero matrix. A = [[3, −7], [−1, 5]]
Step 1: Write down the original matrix A.
A=[3−7−15]A = \begin{bmatrix} 3 & -7 \\ -1 & 5 \end{bmatrix}
Step 2: Negate every element to form −A. Positive values become negative, negative values become positive.
−A=[−371−5]-A = \begin{bmatrix} -3 & 7 \\ 1 & -5 \end{bmatrix}
Step 3: Add A and −A by adding corresponding elements.
A+(−A)=[3+(−3)−7+7−1+15+(−5)]=[0000]A + (-A) = \begin{bmatrix} 3 + (-3) & -7 + 7 \\ -1 + 1 & 5 + (-5) \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}
Step 4: Confirm the result is the 2 × 2 zero matrix, verifying our answer.
A+(−A)=OA + (-A) = O
Answer: The additive inverse is −A = [[-3, 7], [1, -5]], and A + (−A) equals the zero matrix.

Another Example

This example uses a larger 3 × 3 matrix and includes zero elements to show that the additive inverse of 0 is still 0.

Problem: Find the additive inverse of the 3 × 3 matrix B. B = [[0, −4, 2], [6, 0, −1], [−3, 8, 0]]
Step 1: Write down the original matrix B.
B=[0−4260−1−380]B = \begin{bmatrix} 0 & -4 & 2 \\ 6 & 0 & -1 \\ -3 & 8 & 0 \end{bmatrix}
Step 2: Negate every element. Note that the negation of 0 is still 0.
−B=[04−2−6013−80]-B = \begin{bmatrix} 0 & 4 & -2 \\ -6 & 0 & 1 \\ 3 & -8 & 0 \end{bmatrix}
Step 3: Verify by adding B and −B element by element.
B+(−B)=[000000000]=OB + (-B) = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = O
Answer: The additive inverse is −B = [[0, 4, −2], [−6, 0, 1], [3, −8, 0]].

Frequently Asked Questions

What is the difference between the additive inverse and the multiplicative inverse of a matrix?
The additive inverse of A is −A, found by negating every element; adding it to A gives the zero matrix. The multiplicative inverse (also called the inverse matrix, written A⁻¹) is the matrix such that A · A⁻¹ = I, the identity matrix. Not every matrix has a multiplicative inverse, but every matrix has an additive inverse.
Does the additive inverse of a matrix always exist?
Yes. Every matrix, regardless of its size or the values of its elements, has an additive inverse. You simply negate each element. This is guaranteed because every real number has an additive inverse.
How is finding the additive inverse of a matrix related to matrix subtraction?
Matrix subtraction is defined as adding the additive inverse: A − B = A + (−B). So when you subtract one matrix from another, you are really negating the second matrix and then performing matrix addition.

Additive Inverse (−A) vs. Multiplicative Inverse (A⁻¹)

Additive Inverse (−A)Multiplicative Inverse (A⁻¹)
DefinitionMatrix with every element negatedMatrix such that A · A⁻¹ = I (identity matrix)
FormulaA + (−A) = O (zero matrix)A · A⁻¹ = A⁻¹ · A = I
Always exists?Yes, for every matrix of any sizeNo — only for square matrices with nonzero determinant
How to find itNegate each elementUse row reduction, cofactors, or a formula (e.g., for 2×2 matrices)
Result when combined with AZero matrix OIdentity matrix I

Why It Matters

The additive inverse is fundamental to matrix subtraction, which is defined as A − B = A + (−B). You encounter it frequently when solving systems of linear equations using matrix methods, such as Gaussian elimination, where you add a negated row to another row. Understanding this concept also reinforces the algebraic structure of matrices, which carries into linear algebra courses in college.

Common Mistakes

Mistake: Forgetting to negate zero elements, or changing 0 to some other value.
Correction: The negation of 0 is 0. When forming −A, every element is negated, and −0 = 0, so zeros stay as zeros.
Mistake: Confusing the additive inverse (−A) with the multiplicative inverse (A⁻¹).
Correction: These are completely different operations. The additive inverse just negates each entry and always exists. The multiplicative inverse involves finding a matrix that multiplies with A to produce the identity matrix, and it may not exist.

Related Terms