Questions: x=3(1 3; -2 -1; 3 2)-2(4 6 1; 5 7 2)^t

Transcript text: $x=3 \cdot\left(\begin{array}{rr}1 & 3 \\ -2 & -1 \\ 3 & 2\end{array}\right)-2 \cdot\left(\begin{array}{lll}4 & 6 & 1 \\ 5 & 7 & 2\end{array}\right)^{t}$

Solution

Solution Steps

To solve the given matrix expression, we need to perform matrix operations. First, multiply the first matrix by 3. Then, transpose the second matrix and multiply it by 2. Finally, subtract the second result from the first.

Step 1: Multiply the First Matrix by 3

Given the matrix $ A = \begin{pmatrix} 1 & 3 \\ -2 & -1 \\ 3 & 2 \end{pmatrix} $, we multiply each element by 3:

\[ 3 \cdot A = 3 \cdot \begin{pmatrix} 1 & 3 \\ -2 & -1 \\ 3 & 2 \end{pmatrix} = \begin{pmatrix} 3 & 9 \\ -6 & -3 \\ 9 & 6 \end{pmatrix} \]

Step 2: Transpose and Multiply the Second Matrix by 2

Given the matrix $ B = \begin{pmatrix} 4 & 6 & 1 \\ 5 & 7 & 2 \end{pmatrix} $, we first find its transpose:

\[ B^T = \begin{pmatrix} 4 & 5 \\ 6 & 7 \\ 1 & 2 \end{pmatrix} \]

Then, multiply each element of the transposed matrix by 2:

\[ 2 \cdot B^T = 2 \cdot \begin{pmatrix} 4 & 5 \\ 6 & 7 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 8 & 10 \\ 12 & 14 \\ 2 & 4 \end{pmatrix} \]

Step 3: Subtract the Second Result from the First

Subtract the matrix obtained in Step 2 from the matrix obtained in Step 1:

\[ \begin{pmatrix} 3 & 9 \\ -6 & -3 \\ 9 & 6 \end{pmatrix} - \begin{pmatrix} 8 & 10 \\ 12 & 14 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix} 3 - 8 & 9 - 10 \\ -6 - 12 & -3 - 14 \\ 9 - 2 & 6 - 4 \end{pmatrix} = \begin{pmatrix} -5 & -1 \\ -18 & -17 \\ 7 & 2 \end{pmatrix} \]