Problem 21
Question
Let \(A\) be an \(m \times n\) matrix, \(B\) a \(p \times q\) matrix, and \(C\) an \(r \times s\) matrix. Under what conditions is each defined? Find the size of each when defined. (Note: \(A^{2}\) means \(A A .\) ) $$A(B C)$$
Step-by-Step Solution
Verified Answer
The product \(A(BC)\) is defined when \(q = r\) and \(n = p\). When these conditions are satisfied, the size of the resulting matrix is \((m \times s)\).
1Step 1: In order for the product \(BC\) to be defined, the number of columns of \(B\) must be equal to the number of rows of \(C\). So, we must have: $$q = r$$ #Step 2: Calculate the size of the product \(BC\) when defined#
When the condition from Step 1 is met, the product \(BC\) will be a matrix of size \((p \times s)\). This is because the number of rows in the product comes from the first matrix (matrix \(B\)), and the number of columns in the product comes from the second matrix (matrix \(C\)).
#Step 3: Determine the condition for the product \(A(BC)\)#
2Step 2: Now we need to determine when the product \(A(BC)\) is defined. In order for this multiplication to be defined, the number of columns of \(A\) must be equal to the number of rows of \((BC)\). Since \((BC)\) has \(p\) rows, we must have: $$n = p$$ #Step 4: Calculate the size of the product \(A(BC)\) when defined#
When both conditions from Steps 1 and 3 are met (\(q = r\) and \(n = p\)), the product \(A(BC)\) will be a matrix of size \((m \times s)\). This is because the number of rows in the product comes from the first matrix (matrix \(A\)), and the number of columns in the product comes from the second matrix (matrix \((BC)\)).
In conclusion, the product \(A(BC)\) is defined if \(q = r\) and \(n = p\). When these conditions are met, the size of the resulting matrix is \((m \times s)\).
Key Concepts
Matrix DimensionsMultiplication ConditionsResultant Matrix Size
Matrix Dimensions
In mathematics, understanding the dimensions of a matrix is fundamental before performing operations like addition and multiplication. Each matrix has specific dimensions, defined as rows (m) by columns (n). These dimensions • give insight into the structure of the matrix • determine how matrices can be manipulated • play a crucial role in operations involving matrices
For example, if you have matrix \( A \) with dimensions \( m \times n \), this means \( A \) has \( m \) rows and \( n \) columns.
When dealing with matrices, always note: - Arrange the order as rows by columns.- Ensure to match correct dimensions when planning matrix operations.
Dimensions are integral because they dictate conditions for matrix operations and help calculate the resultant size after these operations.
For example, if you have matrix \( A \) with dimensions \( m \times n \), this means \( A \) has \( m \) rows and \( n \) columns.
When dealing with matrices, always note: - Arrange the order as rows by columns.- Ensure to match correct dimensions when planning matrix operations.
Dimensions are integral because they dictate conditions for matrix operations and help calculate the resultant size after these operations.
Multiplication Conditions
Matrix multiplication is a specific process that requires particular conditions to be met. Not all matrices can simply be multiplied by another. Before anything, you should make sure of certain multiplication conditions:
• **Columns of the first = Rows of the second**: This is crucial. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second. For example, if matrix \( B \) is \( p \times q \) and matrix \( C \) is \( r \times s \), you can multiply those matrices only if \( q = r \).
• Once these conditions are met, multiplication is possible. These conditions ensure that multiplication leads to meaningful results. Understanding and applying them is essential for success in any matrix operation.
• **Columns of the first = Rows of the second**: This is crucial. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second. For example, if matrix \( B \) is \( p \times q \) and matrix \( C \) is \( r \times s \), you can multiply those matrices only if \( q = r \).
• Once these conditions are met, multiplication is possible. These conditions ensure that multiplication leads to meaningful results. Understanding and applying them is essential for success in any matrix operation.
Resultant Matrix Size
When two matrices \( A \) and \( B \) are multiplied respecting their dimensions, the product has a resultant matrix size.
• **Rows from the first matrix and columns from the second:** When multiplying two matrices, the resulting matrix gets its number of rows from the first matrix and the number of columns from the second matrix.
• **Rows from the first matrix and columns from the second:** When multiplying two matrices, the resulting matrix gets its number of rows from the first matrix and the number of columns from the second matrix.
- For example, if \( B \) is \( p \times q \) and \( C \) is \( r \times s \), once verified that \( q = r \), \( B \times C \) results in a matrix of size \( p \times s \).
- When extending this to a product like \( A(BC) \), where \( n = p \), the final result will be a matrix of size \( m \times s \).
Other exercises in this chapter
Problem 21
Let \(g: \mathrm{ASCII} \rightarrow A\) defined by \(g(c)=n,\) where \(A=\\{32,33, \ldots, 126\\}\) and \(n\) denotes the ordinal number of the character \(c .\
View solution Problem 21
Determine if each is true or false. Using Exercise 19 and the identity \((i+1)^{2}-i^{2}=2 i+1,\) find a formula for \(\sum_{i=1}^{n} i\)
View solution Problem 22
Determine if the given function is invertible. If it is not invertible, explain why. \(f: \Sigma^{*} \rightarrow \mathbf{W}\) defined by \(f(x)=\) decimal value
View solution Problem 22
Let \(g: \mathrm{ASCII} \rightarrow A\) defined by \(g(c)=n,\) where \(A=\\{32,33, \ldots, 126\\}\) and \(n\) denotes the ordinal number of the character \(c .\
View solution