Answer :
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].
