Q.The negative of a matrix is obtained by multiplying it by _________.

The key idea is that the negative of a matrix is simply the matrix multiplied by the scalar $-1$.

Step 1: For any matrix $A$, its negative is denoted $-A$.

Step 2: By definition, each element of $-A$ is the negative of the corresponding element of $A$: $(-A)_{ij} = -a_{ij}$. …

The idea is simple: just as the negative of a number $5$ is $-5$ (which is $5 \times (-1)$), the negative of a matrix $A$ is $-A$, defined as $(-1) \cdot A$. Every entry $a_{ij}$ in $A$ becomes $-a_{ij}$ in $-A$.

1. Scalar multiplication definition: For any matrix $A = [a_{ij}]$ and any scalar $k$, the product $kA$ is the matrix where each entry is $k \cdot a_{ij}$.

   So if $k = -1$, then $(-1)A$ has entries $(-1) \cdot a_{ij} = -a_{ij}$.

2. Notation: The negative of $A$ is written as $-A$. It is the additive inverse of $A$ because $A + (-A) = 0$ (the zero matrix).

3. Example: If $A = \begin{bmatrix} 2 & -3 \\ 0 & 5 \end{bmatrix}$, then $-A = \begin{bmatrix} -2 & 3 \\ 0 & -5 \end{bmatrix}$. …

Method: Relate the negative of a matrix to scalar multiplication

Use this for questions about the additive inverse (negative) of a matrix.

Steps

Step 1: Recall that scalar multiplication is entrywise.

For a scalar $k$, $kA$ multiplies every entry: $(kA)_{ij} = k\,a_{ij}$.

Step 2: Identify the scalar that flips every sign. …