Problem: Simplify the matrix addition equations.
The task involves adding two matrices together. To add two matrices, you simply add the corresponding elements of the matrices. Both matrices must have the same dimensions (same number of rows and columns).
####
Step-by-Step Solution:
---
Problem 1:
$$
\begin{bmatrix}
3 & 4 \\
2 & -1 \\
1 & 0
\end{bmatrix}
+
\begin{bmatrix}
9 & 0 \\
1 & -2 \\
5 & 8
\end{bmatrix}
$$
Solution:
1. Add the corresponding elements:
- First row, first column: \(3 + 9 = 12\)
- First row, second column: \(4 + 0 = 4\)
- Second row, first column: \(2 + 1 = 3\)
- Second row, second column: \(-1 + (-2) = -3\)
- Third row, first column: \(1 + 5 = 6\)
- Third row, second column: \(0 + 8 = 8\)
2. The resulting matrix is:
$$
\begin{bmatrix}
12 & 4 \\
3 & -3 \\
6 & 8
\end{bmatrix}
$$
---
Problem 2:
$$
\begin{bmatrix}
5 & 2 \\
-3 & 6 \\
0 & 5
\end{bmatrix}
+
\begin{bmatrix}
2 & 4 \\
-2 & -6 \\
-1 & 9
\end{bmatrix}
$$
Solution:
1. Add the corresponding elements:
- First row, first column: \(5 + 2 = 7\)
- First row, second column: \(2 + 4 = 6\)
- Second row, first column: \(-3 + (-2) = -5\)
- Second row, second column: \(6 + (-6) = 0\)
- Third row, first column: \(0 + (-1) = -1\)
- Third row, second column: \(5 + 9 = 14\)
2. The resulting matrix is:
$$
\begin{bmatrix}
7 & 6 \\
-5 & 0 \\
-1 & 14
\end{bmatrix}
$$
---
Problem 3:
$$
\begin{bmatrix}
2 & 1 \\
-7 & 3 \\
9 & 1
\end{bmatrix}
+
\begin{bmatrix}
0 & 3 \\
5 & -3 \\
6 & 8
\end{bmatrix}
$$
Solution:
1. Add the corresponding elements:
- First row, first column: \(2 + 0 = 2\)
- First row, second column: \(1 + 3 = 4\)
- Second row, first column: \(-7 + 5 = -2\)
- Second row, second column: \(3 + (-3) = 0\)
- Third row, first column: \(9 + 6 = 15\)
- Third row, second column: \(1 + 8 = 9\)
2. The resulting matrix is:
$$
\begin{bmatrix}
2 & 4 \\
-2 & 0 \\
15 & 9
\end{bmatrix}
$$
---
Problem 4:
$$
\begin{bmatrix}
4 & -2 \\
-6 & -2 \\
6 & -7
\end{bmatrix}
+
\begin{bmatrix}
6 & -2 \\
-9 & 3 \\
7 & -1
\end{bmatrix}
$$
Solution:
1. Add the corresponding elements:
- First row, first column: \(4 + 6 = 10\)
- First row, second column: \(-2 + (-2) = -4\)
- Second row, first column: \(-6 + (-9) = -15\)
- Second row, second column: \(-2 + 3 = 1\)
- Third row, first column: \(6 + 7 = 13\)
- Third row, second column: \(-7 + (-1) = -8\)
2. The resulting matrix is:
$$
\begin{bmatrix}
10 & -4 \\
-15 & 1 \\
13 & -8
\end{bmatrix}
$$
---
Final Answers:
1. $$
\boxed{
\begin{bmatrix}
12 & 4 \\
3 & -3 \\
6 & 8
\end{bmatrix}
}
$$
2. $$
\boxed{
\begin{bmatrix}
7 & 6 \\
-5 & 0 \\
-1 & 14
\end{bmatrix}
}
$$
3. $$
\boxed{
\begin{bmatrix}
2 & 4 \\
-2 & 0 \\
15 & 9
\end{bmatrix}
}
$$
4. $$
\boxed{
\begin{bmatrix}
10 & -4 \\
-15 & 1 \\
13 & -8
\end{bmatrix}
}
$$
Parent Tip: Review the logic above to help your child master the concept of matrix operations worksheets.
