If `A=[ x y z],B=[(a,h,g),(h,b,f),(g ,f,c)],C=[alpha beta gamma]^T` then `ABC` is

Options

A. Not defined
B. Is a `3xx3` matrix
C. Is a `1xx1` matrix
D. Is a `3xx2` matrix

Answer

To find the order of the product of matrices \( ABC \), we need to analyze the orders of each matrix involved.

1. **Identify the order of each matrix:**
- Matrix \( A \) is given as \( A = [x \quad y \quad z] \). This is a row matrix with 1 row and 3 columns. Therefore, the order of \( A \) is \( 1 \times 3 \).
- Matrix \( B \) is given as \( B = \begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix} \). This is a square matrix with 3 rows and 3 columns. Therefore, the order of \( B \) is \( 3 \times 3 \).
- Matrix \( C \) is given as \( C = \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} \). This is a column matrix with 3 rows and 1 column. Therefore, the order of \( C \) is \( 3 \times 1 \).

2. **Check the possibility of matrix multiplication:**
- To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix.
- First, we will multiply \( A \) and \( B \):
- The order of \( A \) is \( 1 \times 3 \) and the order of \( B \) is \( 3 \times 3 \).
- Since the number of columns in \( A \) (3) equals the number of rows in \( B \) (3), the multiplication \( AB \) is possible.
- The resulting order of \( AB \) will be \( 1 \times 3 \) (the number of rows from \( A \) and the number of columns from \( B \)).

- Next, we will multiply \( AB \) with \( C \):
- The order of \( AB \) is \( 1 \times 3 \) and the order of \( C \) is \( 3 \times 1 \).
- Again, the number of columns in \( AB \) (3) equals the number of rows in \( C \) (3), so the multiplication \( ABC \) is possible.
- The resulting order of \( ABC \) will be \( 1 \times 1 \) (the number of rows from \( AB \) and the number of columns from \( C \)).

3. **Final Result:**
- Therefore, the order of the product \( ABC \) is \( 1 \times 1 \).

### Summary:
The order of the matrix product \( ABC \) is \( 1 \times 1 \).