Certainly! Let's solve the matrix equation step-by-step.
We have:
[tex]\[
A + \begin{pmatrix}
8 \\
-12 \\
3
\end{pmatrix} = \begin{pmatrix}
0 \\
18 \\
-21
\end{pmatrix}
\][/tex]
Our goal is to solve for matrix [tex]\( A \)[/tex].
### Step 1: Identify the matrix notation
To simplify, we can represent the terms:
- Matrix [tex]\( A \)[/tex] as [tex]\( \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \)[/tex]
- Given matrix [tex]\( B \)[/tex] as [tex]\( \begin{pmatrix} 8 \\ -12 \\ 3 \end{pmatrix} \)[/tex]
- Resulting matrix after addition as [tex]\( \begin{pmatrix} 0 \\ 18 \\ -21 \end{pmatrix} \)[/tex]
Thus, the equation can be decomposed into:
[tex]\[
\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + \begin{pmatrix} 8 \\ -12 \\ 3 \end{pmatrix} = \begin{pmatrix} 0 \\ 18 \\ -21 \end{pmatrix}
\][/tex]
### Step 2: Isolate matrix [tex]\( A \)[/tex]
To find [tex]\( A \)[/tex], we subtract [tex]\( \begin{pmatrix} 8 \\ -12 \\ 3 \end{pmatrix} \)[/tex] from both sides of the equation:
[tex]\[
\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 18 \\ -21 \end{pmatrix} - \begin{pmatrix} 8 \\ -12 \\ 3 \end{pmatrix}
\][/tex]
### Step 3: Perform the subtraction component-wise
Now, subtract the corresponding elements of the matrices:
[tex]\[
a_1 = 0 - 8 = -8
\][/tex]
[tex]\[
a_2 = 18 - (-12) = 18 + 12 = 30
\][/tex]
[tex]\[
a_3 = -21 - 3 = -21 - 3 = -24
\][/tex]
### Step 4: Present the solution matrix [tex]\( A \)[/tex]
Thus, matrix [tex]\( A \)[/tex] is:
[tex]\[
A = \begin{pmatrix}
-8 \\
30 \\
-24
\end{pmatrix}
\][/tex]
So the final solution to the matrix equation is:
[tex]\[
A = \begin{pmatrix}
-8 \\
30 \\
-24
\end{pmatrix}
\][/tex]
