Problem Analysis and Solution
The image contains a set of matrix-related questions. Let's solve each problem step by step.
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Question 1: Subtract the matrices
Subtract the following matrices:
\[
\begin{bmatrix}
-2 \\
0 \\
3 \\
6
\end{bmatrix}
-
\begin{bmatrix}
7 \\
-8 \\
0 \\
4
\end{bmatrix}
\]
Solution:
To subtract two matrices, we subtract the corresponding elements of the matrices. The given matrices are column vectors with the same dimensions (4 × 1). Perform the subtraction element-wise:
\[
\begin{bmatrix}
-2 \\
0 \\
3 \\
6
\end{bmatrix}
-
\begin{bmatrix}
7 \\
-8 \\
0 \\
4
\end{bmatrix}
=
\begin{bmatrix}
-2 - 7 \\
0 - (-8) \\
3 - 0 \\
6 - 4
\end{bmatrix}
=
\begin{bmatrix}
-9 \\
8 \\
3 \\
2
\end{bmatrix}
\]
Answer:
The resulting matrix is:
\[
\begin{bmatrix}
-9 \\
8 \\
3 \\
2
\end{bmatrix}
\]
Thus, the correct option is:
\[
\boxed{B}
\]
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####
Question 2: Identify the element in the second row, third column of Matrix R
Matrix \( R \) is given as:
\[
R =
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9.01
\end{bmatrix}
\]
What is the element in the second row, third column?
Solution:
To find the element in the second row, third column, locate the position (2, 3) in the matrix:
- The second row is: \( [4, 5, 6] \).
- The third column is the last column in this row.
The element at position (2, 3) is \( 6 \).
Answer:
The element in the second row, third column is:
\[
\boxed{D}
\]
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####
Question 3: Conditions for adding two matrices
What must be true in order to add two matrices?
Solution:
To add two matrices, the following condition must be satisfied:
- The
dimensions of the matrices must be equal. This means both matrices must have the same number of rows and the same number of columns.
Other options are incorrect because:
- The determinant being non-zero is irrelevant for addition (it matters for invertibility).
- Matrices do not need to be square to be added.
- The number of columns in the first matrix does not need to equal the number of rows in the second matrix (this condition is relevant for matrix multiplication, not addition).
Answer:
The correct condition is:
\[
\boxed{C}
\]
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####
Question 4: Add the matrices
Add the following matrices:
\[
\begin{bmatrix}
4 & 5 \\
1 & -2
\end{bmatrix}
+
\begin{bmatrix}
7 & 3 \\
-5 & 4
\end{bmatrix}
\]
Solution:
To add two matrices, we add the corresponding elements of the matrices. The given matrices are both 2 × 2, so they can be added. Perform the addition element-wise:
\[
\begin{bmatrix}
4 & 5 \\
1 & -2
\end{bmatrix}
+
\begin{bmatrix}
7 & 3 \\
-5 & 4
\end{bmatrix}
=
\begin{bmatrix}
4 + 7 & 5 + 3 \\
1 + (-5) & -2 + 4
\end{bmatrix}
=
\begin{bmatrix}
11 & 8 \\
-4 & 2
\end{bmatrix}
\]
Answer:
The resulting matrix is:
\[
\begin{bmatrix}
11 & 8 \\
-4 & 2
\end{bmatrix}
\]
Thus, the correct answer is:
\[
\boxed{A}
\]
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Final Answers:
1. \(\boxed{B}\)
2. \(\boxed{D}\)
3. \(\boxed{C}\)
4. \(\boxed{A}\)
Parent Tip: Review the logic above to help your child master the concept of 12 grade math worksheet.
