Additive Inverse of a matrix

High School

Definition

Additive inverse rule applied to a matrix. The rule will change the sign of every matrix element from positive (+) to negative (-) or vice versa. The sum of the original matrix and its additive inverse matrix would result in a matrix full of zeros, or "zero matrix."

Worked examples

\(A = \begin{bmatrix} 3 & -2 \)
\( 1 & 5 \end{bmatrix}, \quad -A = \begin{bmatrix} -3 & 2 \)
\( -1 & -5 \end{bmatrix}\)

Every element flips sign; positive becomes negative and negative becomes positive.

\(A + (-A) = \begin{bmatrix} 3 & -2 \)
\( 1 & 5 \end{bmatrix} + \begin{bmatrix} -3 & 2 \)
\( -1 & -5 \end{bmatrix} = \begin{bmatrix} 0 & 0 \)
\( 0 & 0 \end{bmatrix}\)

Adding a matrix and its additive inverse always produces the zero matrix.

Common mistakes

\(-\begin{bmatrix} 2 & -3 \)
\( 4 & 1 \end{bmatrix} = \begin{bmatrix} -2 & -3 \)
\( -4 & -1 \end{bmatrix}\) → \(-\begin{bmatrix} 2 & -3 \)
\( 4 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 3 \)
\( -4 & -1 \end{bmatrix}\) Flip every element's sign, including negatives; \(-(-3) = 3\).
\(A - B = A + B\) → \(A - B = A + (-B)\) Subtracting a matrix means adding its additive inverse.
The additive inverse changes dimension from \(2 \times 2\) to \(2 \times 3\). → The additive inverse has the same dimensions as the original matrix. Only the signs change; the shape stays identical.

Where you'll use it next

You'll use additive inverses when solving matrix equations, performing matrix subtraction, and working with vector spaces in linear algebra and multivariable calculus.

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Matrix Addition and Subtraction: Essential Rules and Techniques

Grade 12 Math

Unlock the power of matrix operations! Learn the fundamental rules for adding and subtracting matrices, practice with step-by-step examples, and avoid common pitfalls in your calculations.

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