Consider the matrices
`A=[(4,6,-1),(3,0,2),(1,-2,5)], B=[(2,4),(0,1),(-1,2)], C=[(3),(1),(2)]`
Out of the given matrix products, which one is not defined ?

Options

A. `(AB)^(T)C`
B. `C^(T)C (AB)^(T)`
C. `C^(T)AB`
D. `A^(T)AB B^(T)C`

Explanation

To determine which of the given matrix products is not defined, we need to analyze the dimensions of the matrices involved in each product. Let's start by identifying the dimensions of the matrices A, B, and C.

1. **Matrix A**:
- A = \(\begin{pmatrix} 4 & 6 & -1 \\ 3 & 0 & 2 \\ 1 & -2 & 5 \end{pmatrix}\)
- Dimensions: 3 rows and 3 columns (3x3)

2. **Matrix B**:
- B = \(\begin{pmatrix} 2 & 4 \\ 0 & 1 \\ -1 & 2 \end{pmatrix}\)
- Dimensions: 3 rows and 2 columns (3x2)

3. **Matrix C**:
- C = \(\begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix}\)
- Dimensions: 3 rows and 1 column (3x1)

Now, let's analyze each option one by one:

### Option A: \( (AB)^T C \)

- First, we calculate \( AB \):
- A (3x3) and B (3x2) can be multiplied because the number of columns in A (3) matches the number of rows in B (3).
- Resulting dimensions of \( AB \) will be (3x2).

- Now, we take the transpose: \( (AB)^T \):
- The transpose of a (3x2) matrix is (2x3).

- Finally, we multiply \( (AB)^T \) (2x3) with C (3x1):
- The multiplication is valid because the number of columns in \( (AB)^T \) (3) matches the number of rows in C (3).
- Resulting dimensions will be (2x1).

**Conclusion**: Option A is defined.

### Option B: \( C^T C AB \)

- First, we calculate \( C^T \):
- C (3x1) transposed becomes (1x3).

- Now, we multiply \( C^T \) (1x3) with C (3x1):
- The multiplication is valid because the number of columns in \( C^T \) (3) matches the number of rows in C (3).
- Resulting dimensions will be (1x1).

- Next, we multiply \( (C^T C) \) (1x1) with \( AB \) (3x2):
- This multiplication is **not valid** because the number of columns in \( (C^T C) \) (1) does not match the number of rows in \( AB \) (3).

**Conclusion**: Option B is not defined.

### Option C: \( C^T AB \)

- We already know \( C^T \) is (1x3) and \( AB \) is (3x2).
- The multiplication is valid because the number of columns in \( C^T \) (3) matches the number of rows in \( AB \) (3).
- Resulting dimensions will be (1x2).

**Conclusion**: Option C is defined.

### Option D: \( A^T AB B^T C \)

- First, we calculate \( A^T \):
- A (3x3) transposed becomes (3x3).

- Next, we multiply \( A^T \) (3x3) with \( AB \) (3x2):
- This multiplication is valid.
- Resulting dimensions will be (3x2).

- Now, we calculate \( B^T \):
- B (3x2) transposed becomes (2x3).

- Next, we multiply \( AB B^T \):
- \( AB \) (3x2) and \( B^T \) (2x3) can be multiplied.
- Resulting dimensions will be (3x3).

- Finally, we multiply \( (A^T AB B^T) \) (3x3) with C (3x1):
- This multiplication is valid.
- Resulting dimensions will be (3x1).

**Conclusion**: Option D is defined.

### Final Conclusion:
Out of the given matrix products, the one that is **not defined** is **Option B**: \( C^T C AB \).