Understand the additive inverse of a matrix.

Step 1: Understand the additive inverse of a matrix. The additive inverse of a matrix is found by negating each element of the matrix. If $A$ is a matrix, its additive inverse is $-A$, such that $A + (-A) = O$, where $O$ is the zero matrix. Step 2: Find the additive inverse of the given matrix. The given matrix is $\begin{bmatrix} -1 & 2 \\ 3 & -4 \end{bmatrix}$. Its additive inverse is: $$ -\begin{bmatrix} -1 & 2 \\ 3 & -4 \end{bmatrix} = \begin{bmatrix} -(-1) & -(2) \\ -(3) & -(-4) \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ -3 & 4 \end{bmatrix} $$ Step 3: Equate the calculated additive inverse with the given additive inverse. We are given that the additive inverse is $\begin{bmatrix} a & -2 \\ -3 & b \end{bmatrix}$. Therefore, we can set up the equality: $$ \begin{bmatrix} a & -2 \\ -3 & b \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ -3 & 4 \end{bmatrix} $$ Step 4: Compare corresponding elements to find the values of $a$ and $b$. By comparing the elements in the same positions, we get: $a = 1$ $b = 4$ The values are $a = 1$ and $b = 4$. The final answer is $\boxed{a=1, b=4}$. Send me the next one 📸