Question

problem 7. (14 points)
let ( c = \begin{bmatrix} 1 & 2 \\ 2 & 2 \\ 6 & 1 end{bmatrix} ) and ( e = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 2 & 1 \\ 4 & 3 & 1 end{bmatrix} ).
find ( ec ). if it is not possible to perform this operation, explain.
select the correct choice below, and, if necessary, fill in the answer boxes to complete your choice.
( \bigcirc ) a. ( ec = \begin{bmatrix} square & square \\ square & square \\ square & square end{bmatrix} )
( \bigcirc ) b. the product of two matrices of different sizes is not defined.
( \bigcirc ) c. the operation is not defined because the number of columns in the first matrix is not the same as the number of rows in the second matrix.
( \bigcirc ) d. the operation is not defined because the number of rows in the first matrix is not the same as the number of columns in the second matrix.

Explanation:

Step1: Determine matrix dimensions

Matrix \( E \) has dimensions \( 3 \times 3 \) (3 rows, 3 columns), matrix \( C \) has dimensions \( 3 \times 2 \) (3 rows, 2 columns).

Step2: Check matrix multiplication rule

For matrix multiplication \( EC \), the number of columns in \( E \) (3) must equal the number of rows in \( C \) (3). Since \( 3 = 3 \), multiplication is defined.

Step3: Perform matrix multiplication

First row of \( E \) times first column of \( C \): \( (1)(1) + (0)(2) + (1)(6) = 1 + 0 + 6 = 7 \)
First row of \( E \) times second column of \( C \): \( (1)(2) + (0)(2) + (1)(1) = 2 + 0 + 1 = 3 \)
Second row of \( E \) times first column of \( C \): \( (1)(1) + (2)(2) + (1)(6) = 1 + 4 + 6 = 11 \)
Second row of \( E \) times second column of \( C \): \( (1)(2) + (2)(2) + (1)(1) = 2 + 4 + 1 = 7 \)
Third row of \( E \) times first column of \( C \): \( (4)(1) + (3)(2) + (1)(6) = 4 + 6 + 6 = 16 \)
Third row of \( E \) times second column of \( C \): \( (4)(2) + (3)(2) + (1)(1) = 8 + 6 + 1 = 15 \)

So \( EC =

$$\begin{bmatrix} 7 & 3 \\ 11 & 7 \\ 16 & 15 \end{bmatrix}$$

\)

Answer:

A. \( EC =

$$\begin{bmatrix} 7 & 3 \\ 11 & 7 \\ 16 & 15 \end{bmatrix}$$

\)