Question

The additive inverse of a matrix A is:
(a) – A
(b) |A|
(c) \[{{A}^{2}}\]
(d) \[\dfrac{adj\text{ }A}{\left| A \right|}\]

Answer 1

Hint: First of all try to recollect the meaning of additive inverse of a number. That is for number n, the additive inverse is – n. Now apply the same rule to find the additive inverse of matrix A.

Complete step-by-step answer:
Here we have to find the additive inverse matrix A. Before proceeding with the question, let us know about the terms used in the question.
Additive Inverse: In mathematics, the additive inverse of a number is the number which when added to it, yields zero. This number is also known as the opposite number, sign change, or negation. For a real number, the additive inverse reverses its sign. The opposite of a positive number is negative and the opposite of a negative number is positive. Zero is the additive inverse of itself. For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of − 0.3 is 0.3, because −0.3 + 0.3 = 0. For a number, generally, the additive inverse can be calculated using multiplication by −1.
Matrix: In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions that are arranged in rows and columns. The dimensions of a matrix is denoted by the number of rows and columns. For example, the dimension of the matrix below is 3 × 3 because there are three rows and three columns in it. The dimension of the matrix is read as three by three.
\[\left[ \begin{matrix}
   5 & 8 & 0 \\
   6 & 1 & 12 \\
   7 & 2 & 5 \\
\end{matrix} \right]\]
Additive inverse of Matrix: The matrix obtained by changing the sign of every matrix element is known as the additive inverse of the matrix. Suppose we have a matrix,
\[M=\left[ \begin{matrix}
   -5 & 8 & -9 \\
   6 & -1 & 12 \\
   -7 & 2 & 5 \\
\end{matrix} \right]\]
Then, we get its inverse by changing the sign of every element. So we get, the additive inverse of the matrix as,
\[M=\left[ \begin{matrix}
   5 & -8 & 9 \\
   -6 & 1 & -12 \\
   7 & -2 & -5 \\
\end{matrix} \right]\]
We know that we can write it as,
\[\left[ \begin{matrix}
   5 & -8 & 9 \\
   -6 & 1 & -12 \\
   7 & -2 & -5 \\
\end{matrix} \right]=-M\]
So, we get the additive inverse of matrix M = – M. Now let us consider our question, we have to find the additive inverse matrix A. As we have already seen that additive inverse of a matrix is the matrix obtained by changing the sign of every matrix element.
So we get, additive inverse of matrix A = – A.
Hence, option (a) is the correct answer.

Note: Some students consider the additive inverse of matrix and inverse of a matrix to be the same which is wrong. Additive inverse of any matrix M is – M while the inverse of any matrix M is \[{{\left( M \right)}^{-1}}\] and both are very different. Also, students must note that the sum of a matrix and its additive inverse is always a zero matrix. For example, the sum of matrix A and its additive inverse – A is A + (– A) = A – A = 0 (zero matrix).