[2001] [0 2 1]^T [5 1 6] = ( )

Steps to Solve

1. Identify the matrices The matrices given are: $$ A = \begin{bmatrix} 2 & 0 \ 0 & 2 \ 1 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 5 \ 1 \ 6 \end{bmatrix} $$

2. Determine the dimensions Matrix A is a (3 \times 2) matrix and matrix B is a (3 \times 1) vector.

3. Matrix multiplication setup To find the product ( AB ), we need to ensure the inner dimensions match. Since A has (2) columns and B has (3) rows, we can't multiply them directly.

4. Transpose matrix B Transposing B gives us a (1 \times 3) matrix: $$ B^T = \begin{bmatrix} 5 & 1 & 6 \end{bmatrix} $$

5. Perform the multiplication Now, multiply matrix A by the transposed matrix B: $$ AB^T = \begin{bmatrix} 2 & 0 \ 0 & 2 \ 1 & 6 \end{bmatrix} \begin{bmatrix} 5 \ 1 \ 6 \end{bmatrix} $$

6. Calculate each entry of the resulting matrix For each element in the resulting matrix:

• First row: ( (2 \cdot 5) + (0 \cdot 1) + (0 \cdot 6) = 10 )
• Second row: ( (0 \cdot 5) + (2 \cdot 1) + (0 \cdot 6) = 2 )
• Third row: ( (1 \cdot 5) + (0 \cdot 1) + (6 \cdot 6) = 5 + 0 + 36 = 41 )

7. Form the result matrix The resulting matrix after multiplication is: $$ \begin{bmatrix} 10 \ 2 \ 41 \end{bmatrix} $$