Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
Let's break down the problem step-by-step.
Given:
- [tex]\( A \)[/tex] is a [tex]\( 4 \times 1 \)[/tex] matrix.
- [tex]\( B \)[/tex] is a [tex]\( 4 \times 5 \)[/tex] matrix.
- [tex]\( C \)[/tex] is a [tex]\( 3 \times 3 \)[/tex] matrix.
- [tex]\( D \)[/tex] is a [tex]\( 3 \times 5 \)[/tex] matrix.
We need to analyze the expression:
[tex]\[ A^t \left(D^t B\right)^t - \alpha B \left( C D^t \right)^t \][/tex]
### Step 1: Transpose Matrices
First, calculate the shapes of the transposed matrices:
1. [tex]\( A^t \)[/tex]: If [tex]\( A \)[/tex] is [tex]\( 4 \times 1 \)[/tex], then [tex]\( A^t \)[/tex] will be [tex]\( 1 \times 4 \)[/tex].
2. [tex]\( D^t \)[/tex]: If [tex]\( D \)[/tex] is [tex]\( 3 \times 5 \)[/tex], then [tex]\( D^t \)[/tex] will be [tex]\( 5 \times 3 \)[/tex].
### Step 2: Determine if Intermediate Products are Defined and Their Shapes
1. Intermediate Product [tex]\( D^t B \)[/tex]:
- To multiply [tex]\( D^t \)[/tex] by [tex]\( B \)[/tex], the inner dimensions must match.
- [tex]\( D^t \)[/tex] has dimensions [tex]\( 5 \times 3 \)[/tex] and [tex]\( B \)[/tex] has dimensions [tex]\( 4 \times 5 \)[/tex].
- Since the inner dimensions (3 and 4) do not match, [tex]\( D^t B \)[/tex] is not defined.
2. Intermediate Product [tex]\( C D^t \)[/tex]:
- To multiply [tex]\( C \)[/tex] by [tex]\( D^t \)[/tex], the inner dimensions must match.
- [tex]\( C \)[/tex] has dimensions [tex]\( 3 \times 3 \)[/tex] and [tex]\( D^t \)[/tex] has dimensions [tex]\( 5 \times 3 \)[/tex].
- Since the inner dimensions (3 and 5) do not match, [tex]\( C D^t \)[/tex] is not defined.
### Step 3: Analyze the Full Expression
Since neither [tex]\( D^t B \)[/tex] nor [tex]\( C D^t \)[/tex] is defined, any further operations involving these intermediate products are also not defined.
### Step 4: Conclusion
The matrix expression [tex]\( A^t \left(D^t B\right)^t - \alpha B \left(C D^t\right)^t \)[/tex] is not defined because both [tex]\( D^t B \)[/tex] and [tex]\( C D^t \)[/tex] are not defined due to incompatible dimensions.
### Example Matrices
Here are example matrices [tex]\( A, B, C, \)[/tex] and [tex]\( D \)[/tex] such that each product [tex]\( D^t B \)[/tex] and [tex]\( C D^t \)[/tex] is not defined:
- [tex]\( A = \begin{pmatrix} 0.24823848 \\ 0.78088306 \\ 0.88798074 \\ 0.30060036 \end{pmatrix} \)[/tex]
- [tex]\( B = \begin{pmatrix} 0.31593352 & 0.78191155 & 0.83340476 & 0.00370335 & 0.18508896 \\ 0.83815913 & 0.98142136 & 0.44767318 & 0.71397522 & 0.37753253 \\ 0.16569744 & 0.3771317 & 0.55552327 & 0.09828603 & 0.94650052 \\ 0.4925713 & 0.34788751 & 0.81792057 & 0.01453707 & 0.00103607 \end{pmatrix} \)[/tex]
- [tex]\( C = \begin{pmatrix} 0.75958717 & 0.63391352 & 0.76282272 \\ 0.08237095 & 0.50743485 & 0.03009981 \\ 0.09947913 & 0.6503981 & 0.56393987 \end{pmatrix} \)[/tex]
- [tex]\( D = \begin{pmatrix} 0.50054946 & 0.21858296 & 0.66804206 & 0.51062889 & 0.59322394 \\ 0.80024769 & 0.07609848 & 0.53716718 & 0.72575023 & 0.96149933 \\ 0.61313327 & 0.49820059 & 0.3388521 & 0.61316237 & 0.91819739 \end{pmatrix} \)[/tex]
In summary, the given matrix expression is not defined due to dimensional mismatches in the intermediate products [tex]\( D^t B \)[/tex] and [tex]\( C D^t \)[/tex].
Given:
- [tex]\( A \)[/tex] is a [tex]\( 4 \times 1 \)[/tex] matrix.
- [tex]\( B \)[/tex] is a [tex]\( 4 \times 5 \)[/tex] matrix.
- [tex]\( C \)[/tex] is a [tex]\( 3 \times 3 \)[/tex] matrix.
- [tex]\( D \)[/tex] is a [tex]\( 3 \times 5 \)[/tex] matrix.
We need to analyze the expression:
[tex]\[ A^t \left(D^t B\right)^t - \alpha B \left( C D^t \right)^t \][/tex]
### Step 1: Transpose Matrices
First, calculate the shapes of the transposed matrices:
1. [tex]\( A^t \)[/tex]: If [tex]\( A \)[/tex] is [tex]\( 4 \times 1 \)[/tex], then [tex]\( A^t \)[/tex] will be [tex]\( 1 \times 4 \)[/tex].
2. [tex]\( D^t \)[/tex]: If [tex]\( D \)[/tex] is [tex]\( 3 \times 5 \)[/tex], then [tex]\( D^t \)[/tex] will be [tex]\( 5 \times 3 \)[/tex].
### Step 2: Determine if Intermediate Products are Defined and Their Shapes
1. Intermediate Product [tex]\( D^t B \)[/tex]:
- To multiply [tex]\( D^t \)[/tex] by [tex]\( B \)[/tex], the inner dimensions must match.
- [tex]\( D^t \)[/tex] has dimensions [tex]\( 5 \times 3 \)[/tex] and [tex]\( B \)[/tex] has dimensions [tex]\( 4 \times 5 \)[/tex].
- Since the inner dimensions (3 and 4) do not match, [tex]\( D^t B \)[/tex] is not defined.
2. Intermediate Product [tex]\( C D^t \)[/tex]:
- To multiply [tex]\( C \)[/tex] by [tex]\( D^t \)[/tex], the inner dimensions must match.
- [tex]\( C \)[/tex] has dimensions [tex]\( 3 \times 3 \)[/tex] and [tex]\( D^t \)[/tex] has dimensions [tex]\( 5 \times 3 \)[/tex].
- Since the inner dimensions (3 and 5) do not match, [tex]\( C D^t \)[/tex] is not defined.
### Step 3: Analyze the Full Expression
Since neither [tex]\( D^t B \)[/tex] nor [tex]\( C D^t \)[/tex] is defined, any further operations involving these intermediate products are also not defined.
### Step 4: Conclusion
The matrix expression [tex]\( A^t \left(D^t B\right)^t - \alpha B \left(C D^t\right)^t \)[/tex] is not defined because both [tex]\( D^t B \)[/tex] and [tex]\( C D^t \)[/tex] are not defined due to incompatible dimensions.
### Example Matrices
Here are example matrices [tex]\( A, B, C, \)[/tex] and [tex]\( D \)[/tex] such that each product [tex]\( D^t B \)[/tex] and [tex]\( C D^t \)[/tex] is not defined:
- [tex]\( A = \begin{pmatrix} 0.24823848 \\ 0.78088306 \\ 0.88798074 \\ 0.30060036 \end{pmatrix} \)[/tex]
- [tex]\( B = \begin{pmatrix} 0.31593352 & 0.78191155 & 0.83340476 & 0.00370335 & 0.18508896 \\ 0.83815913 & 0.98142136 & 0.44767318 & 0.71397522 & 0.37753253 \\ 0.16569744 & 0.3771317 & 0.55552327 & 0.09828603 & 0.94650052 \\ 0.4925713 & 0.34788751 & 0.81792057 & 0.01453707 & 0.00103607 \end{pmatrix} \)[/tex]
- [tex]\( C = \begin{pmatrix} 0.75958717 & 0.63391352 & 0.76282272 \\ 0.08237095 & 0.50743485 & 0.03009981 \\ 0.09947913 & 0.6503981 & 0.56393987 \end{pmatrix} \)[/tex]
- [tex]\( D = \begin{pmatrix} 0.50054946 & 0.21858296 & 0.66804206 & 0.51062889 & 0.59322394 \\ 0.80024769 & 0.07609848 & 0.53716718 & 0.72575023 & 0.96149933 \\ 0.61313327 & 0.49820059 & 0.3388521 & 0.61316237 & 0.91819739 \end{pmatrix} \)[/tex]
In summary, the given matrix expression is not defined due to dimensional mismatches in the intermediate products [tex]\( D^t B \)[/tex] and [tex]\( C D^t \)[/tex].
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
