Questions: Let A = [1 0 0 0 1 0 1 3 4] = [A11 A12 A21 A22] and B = [1 0 2 0 3 4] = [B11 B12 B21 B22]. (a) Find (i) A11 B11 + A12 B21, (ii) A11 B12 + A12 B22, (iii) A21 B11 + A22 B21, (iv) A21 B12 + A22 B22 (b) Find AB using your work from part (a).

Let A = [1 0 0 0 1 0 1 3 4] = [A11 A12 A21 A22] and B = [1 0 2 0 3 4] = [B11 B12 B21 B22].
(a) Find (i) A11 B11 + A12 B21, (ii) A11 B12 + A12 B22, (iii) A21 B11 + A22 B21, (iv) A21 B12 + A22 B22
(b) Find AB using your work from part (a).

Transcript text: 3. Let $A=\left[\begin{array}{ll|l}1 & 0 & 0 \\ 0 & 1 & 0 \\ \hline 1 & 3 & 4\end{array}\right]=\left[\begin{array}{l|l}A_{11} & A_{12} \\ \hline A_{21} & A_{22}\end{array}\right]$ and $B=\left[\begin{array}{l|l}1 & 0 \\ 2 & 0 \\ \hline 3 & 4\end{array}\right]=\left[\begin{array}{l|l}B_{11} & B_{12} \\ \hline B_{21} & B_{22}\end{array}\right]$. (a) Find (i) $A_{11} B_{11}+A_{12} B_{21}$, (ii) $A_{11} B_{12}+A_{12} B_{22}$, (iii) $A_{21} B_{11}+A_{22} B_{21}$, (iv) $A_{21} B_{12}+A_{22} B_{22}$ (b) Find $A B$ using your work from part (a).

Solution

Solution Steps

To solve this problem, we need to perform matrix multiplication using block matrices. The matrices $ A $ and $ B $ are partitioned into submatrices. We will calculate each of the specified products using the submatrices and then combine these results to find the product $ AB $.

Identify the submatrices $ A_{11}, A_{12}, A_{21}, A_{22} $ from matrix $ A $ and $ B_{11}, B_{12}, B_{21}, B_{22} $ from matrix $ B $.
Compute the products:
- $ A_{11} B_{11} + A_{12} B_{21} $
- $ A_{11} B_{12} + A_{12} B_{22} $
- $ A_{21} B_{11} + A_{22} B_{21} $
- $ A_{21} B_{12} + A_{22} B_{22} $
Use these results to construct the matrix product $ AB $.

Step 1: Identify the Submatrices

The matrices $ A $ and $ B $ are given as follows:

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

We can identify the submatrices:

$ A_{11} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $
$ A_{12} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $
$ A_{21} = \begin{bmatrix} 1 & 3 \end{bmatrix} $
$ A_{22} = \begin{bmatrix} 4 \end{bmatrix} $
$ B_{11} = \begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix} $
$ B_{12} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} $
$ B_{21} = \begin{bmatrix} 3 & 4 \end{bmatrix} $
$ B_{22} = \begin{bmatrix} 0 \end{bmatrix} $

Step 2: Calculate the Required Products

We compute the following products:

$ C_{11} = A_{11} B_{11} + A_{12} B_{21} $ \[ C_{11} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \end{bmatrix} \begin{bmatrix} 3 & 4 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix} \]
$ C_{12} = A_{11} B_{12} + A_{12} B_{22} $ \[ C_{12} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \end{bmatrix} \begin{bmatrix} 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \]
$ C_{21} = A_{21} B_{11} + A_{22} B_{21} $ \[ C_{21} = \begin{bmatrix} 1 & 3 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 2 & 0 \end{bmatrix} + \begin{bmatrix} 4 \end{bmatrix} \begin{bmatrix} 3 & 4 \end{bmatrix} = \begin{bmatrix} 1 + 6 & 0 + 12 \end{bmatrix} = \begin{bmatrix} 7 & 12 \end{bmatrix} \]
$ C_{22} = A_{21} B_{12} + A_{22} B_{22} $ \[ C_{22} = \begin{bmatrix} 1 & 3 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix} + \begin{bmatrix} 4 \end{bmatrix} \begin{bmatrix} 0 \end{bmatrix} = 3 + 12 = 15 \]

Step 3: Construct the Resulting Matrix $ AB $

Now we can construct the resulting matrix $ AB $ using the computed blocks:

\[ AB = \begin{bmatrix} C_{11} & C_{12} \\ C_{21} & C_{22} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 3 \\ 2 & 0 & 4 \\ 19 & 16 & 15 \end{bmatrix} \]