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Chapter 4: Problem 10

Describing the Additive Inverse In Exercises \(7-12\), describe the additive inverse of a vector in the vector space. $$ M_{1,4} $$

Short Answer

Expert verified
The additive inverse of a matrix A in the vector space \(M_{1,4}\) is \(-A\).

Step by step solution

01

Understanding the Additive Inverse

The additive inverse of a matrix (or any vector in a vector space) is a matrix (vector) that, when added to the original matrix (vector), results in the zero matrix (zero vector). In other words, if the original matrix is 'A', its additive inverse 'B' satisfies the condition A + B = 0.
02

Determining the Additive Inverse

Assuming the matrix A in \(M_{1,4}\) to be \( [a, b, c, d] \), the additive inverse of A is thus \(-A\) or \( [-a, -b, -c, -d] \). It can be verified by adding A and \(-A\). This results in the zero matrix \([0,0,0,0]\), confirming that \(-A\) is indeed the additive inverse of A in the vector space \(M_{1,4}\).

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