Problem 7
Question
Let \(A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 0 & 2 & 3\end{array}\right], B=\left[\begin{array}{ccc}0 & -2 & 5 \\ 0 & 0 & 1\end{array}\right],\) and \(C=\left[\begin{array}{ccc}-3 & 0 & 0 \\ 0 & 1 & 2\end{array}\right] .\) Find each. $$B+C$$
Step-by-Step Solution
Verified Answer
The sum of matrices B and C is:
\[
B+C =
\left[\begin{array}{ccc}
-3 & -2 & 5 \\
0 & 1 & 3
\end{array}\right].
\]
1Step 1: Verify Dimensions
Check if the dimensions of B and C are the same. In this case, both matrices are 2x3 matrices. Since they have the same dimensions, we can proceed with matrix addition.
2Step 2: Add Corresponding Elements
To obtain the sum, add the corresponding elements of matrices B and C:
\[(B+C)_{ij} = B_{ij} + C_{ij}\]
Which results in:
\[
B+C =
\left[\begin{array}{ccc}
(0) + (-3)& (-2) + (0) & (5) + (0) \\
(0) + (0) & (0) + (1) & (1) + (2)
\end{array}\right]
\]
3Step 3: Simplify the Result
Simplify the elements within the resulting matrix:
\[
B+C =
\left[\begin{array}{ccc}
-3 & -2 & 5 \\
0 & 1 & 3
\end{array}\right]
\]
So, the sum of matrices B and C is:
\[
B+C =
\left[\begin{array}{ccc}
-3 & -2 & 5 \\
0 & 1 & 3
\end{array}\right].
\]
Key Concepts
Matrix OperationsElement-wise AdditionMatrices DimensionsResulting Matrix Simplification
Matrix Operations
Matrix operations are foundational in many areas of mathematics, science, and engineering. They allow us to manipulate matrices in meaningful ways to uncover new insights or solve complex problems. In the realm of matrix operations, addition is one of the simplest, yet vital techniques. Matrix addition involves adding two matrices of the same dimensions by combining their corresponding elements.
This process is intuitive once understood, as it resembles the traditional addition of numbers. By learning matrix operations, students gain a powerful tool for tackling various computational challenges. Understanding matrix addition as a matrix operation requires ensuring the matrices are compatible in size, highlighting the significance of verifying dimensions before proceeding.
This process is intuitive once understood, as it resembles the traditional addition of numbers. By learning matrix operations, students gain a powerful tool for tackling various computational challenges. Understanding matrix addition as a matrix operation requires ensuring the matrices are compatible in size, highlighting the significance of verifying dimensions before proceeding.
Element-wise Addition
Element-wise addition refers to the process of adding corresponding elements from two matrices. It necessitates that the matrices be of identical dimensions. This is because you cannot add elements if there isn't a corresponding pair.
The formula used is straightforward: given matrices \(A\) and \(B\), the sum \((A+B)_{ij}\) is equal to \(A_{ij} + B_{ij}\). Each element in the resulting matrix is derived from the direct addition of respective elements from the matrices we started with. It's like a column-by-column and row-by-row examination, ensuring that every position is filled with a summation of its matched elements.
This technique is simple but essential for further advancements in more complex matrix operations.
The formula used is straightforward: given matrices \(A\) and \(B\), the sum \((A+B)_{ij}\) is equal to \(A_{ij} + B_{ij}\). Each element in the resulting matrix is derived from the direct addition of respective elements from the matrices we started with. It's like a column-by-column and row-by-row examination, ensuring that every position is filled with a summation of its matched elements.
This technique is simple but essential for further advancements in more complex matrix operations.
Matrices Dimensions
Dimensions of a matrix denote its size, usually represented as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. For any matrix operation, particularly addition, it's crucial to confirm that matrices involved share the same dimensions.
If you're given two matrices - for instance, both being \(2 \times 3\) - they have identical dimensions. Two matrices can only be added if they have the same number of rows and columns. This stipulation ensures that each entry has a counterpart, facilitating element-wise operations.
If you're given two matrices - for instance, both being \(2 \times 3\) - they have identical dimensions. Two matrices can only be added if they have the same number of rows and columns. This stipulation ensures that each entry has a counterpart, facilitating element-wise operations.
- Check row count
- Check column count
Resulting Matrix Simplification
Once matrix elements have been added, the resulting matrix might not represent the final, simplified version right away. Simplifying a matrix involves calculating and expressing each added element in its simplest form.
For example, if an added element yields a computation like \( (0) + (-3) = -3 \), then the straightforward calculation resolves to \(-3\) in the result matrix.
For example, if an added element yields a computation like \( (0) + (-3) = -3 \), then the straightforward calculation resolves to \(-3\) in the result matrix.
- Carry out addition
- Simplify each result
Other exercises in this chapter
Problem 7
Determine if each function is injective, where trunc( \(x\) ) denotes the integral part of the real number of \(x.\) $$f(x)=\lfloor x\rfloor, x \in \mathbb{R}$$
View solution Problem 7
Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x y\rfloor$$
View solution Problem 7
Let \(f(x)=\lfloor x\rfloor\) and \(g(x)\) \(=\lceil x\rceil,\) where \(x \in \mathbb{R} .\) Compute each. $$(g \circ f)(-4.1)$$
View solution Problem 7
Let \(x=3.456\) and \(y=2.789 .\) Compute each. $$\lfloor x y\rfloor$$
View solution