Questions: Use matrices A=[[-1 2] [4 7]], B=[[1 4 -2] [4 -5 0]] and C=[[1 0 -1] [4 0 3] [2 -1 2]] to find AB+BC. AB+BC= (Type an integer or simplified fraction for each matrix element.)

$Use matrices A=[[-1 2] [4 7]], B=[[1 4 -2] [4 -5 0]] and C=[[1 0 -1] [4 0 3] [2 -1 2]] to find AB+BC. AB+BC= (Type an integer or simplified fraction for each matrix element.)$

Transcript text: Use matrices $A=\left[\begin{array}{rr}-1 & 2 \\ 4 & 7\end{array}\right], B=\left[\begin{array}{rrr}1 & 4 & -2 \\ 4 & -5 & 0\end{array}\right]$ and $C=\left[\begin{array}{rrr}1 & 0 & -1 \\ 4 & 0 & 3 \\ 2 & -1 & 2\end{array}\right]$ to find $A B+B C$. \[ A B+B C= \] $\square$ (Type an integer or simplified fraction for each matrix element.)

Solution

Solution Steps

To solve the problem of finding $ AB + BC $, we need to perform matrix multiplication and addition. First, calculate the product of matrices $ A $ and $ B $. Then, calculate the product of matrices $ B $ and $ C $. Finally, add the resulting matrices from the two products to get the final result.

Step 1: Calculate $ AB $

To find the product $ AB $, we perform the matrix multiplication:

\[ A = \begin{bmatrix} -1 & 2 \\ 4 & 7 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 4 & -2 \\ 4 & -5 & 0 \end{bmatrix} \]

Calculating $ AB $:

\[ AB = \begin{bmatrix} (-1)(1) + (2)(4) & (-1)(4) + (2)(-5) & (-1)(-2) + (2)(0) \\ (4)(1) + (7)(4) & (4)(4) + (7)(-5) & (4)(-2) + (7)(0) \end{bmatrix} = \begin{bmatrix} 7 & -14 & 2 \\ 32 & -19 & -8 \end{bmatrix} \]

Step 2: Calculate $ BC $

Next, we calculate the product $ BC $:

\[ B = \begin{bmatrix} 1 & 4 & -2 \\ 4 & -5 & 0 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & -1 \\ 4 & 0 & 3 \\ 2 & -1 & 2 \end{bmatrix} \]

Calculating $ BC $:

\[ BC = \begin{bmatrix} (1)(1) + (4)(4) + (-2)(2) & (1)(0) + (4)(0) + (-2)(-1) & (1)(-1) + (4)(3) + (-2)(2) \\ (4)(1) + (-5)(4) + (0)(2) & (4)(0) + (-5)(0) + (0)(-1) & (4)(-1) + (-5)(3) + (0)(2) \end{bmatrix} = \begin{bmatrix} 13 & 2 & 7 \\ -16 & 0 & -19 \end{bmatrix} \]

Step 3: Calculate $ AB + BC $

Now, we add the results of $ AB $ and $ BC $:

\[ AB = \begin{bmatrix} 7 & -14 & 2 \\ 32 & -19 & -8 \end{bmatrix}, \quad BC = \begin{bmatrix} 13 & 2 & 7 \\ -16 & 0 & -19 \end{bmatrix} \]

Calculating $ AB + BC $:

\[ AB + BC = \begin{bmatrix} 7 + 13 & -14 + 2 & 2 + 7 \\ 32 - 16 & -19 + 0 & -8 - 19 \end{bmatrix} = \begin{bmatrix} 20 & -12 & 9 \\ 16 & -19 & -27 \end{bmatrix} \]