Find the multiplication of the following two matrices. Suppose that F(x, y) means x is the father of y. Write the following statement by using predicate and quantifiers, where the... Find the multiplication of the following two matrices. Suppose that F(x, y) means x is the father of y. Write the following statement by using predicate and quantifiers, where the domain of x and y are all persons in this universe. Father-child relation is not symmetric. Read more

Steps to Solve

1. Identify the matrices

The given matrices are:

$$ A = \begin{pmatrix} 1 & 2 \ 2 & 1 \ \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 2 \ 1 & 1 \ \end{pmatrix} $$

2. Determine the size of the resulting matrix

Matrix $A$ is a $2 \times 2$ matrix and matrix $B$ is also a $2 \times 2$ matrix. The resulting matrix, when multiplying two matrices of size $m \times n$ and $n \times p$, will be a matrix of size $m \times p$. Thus, the resulting matrix will be $2 \times 2$.

3. Perform matrix multiplication

To find the entry in row $i$ and column $j$ of the resulting matrix $C$, we use the formula:

$$ C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj} $$

For our case:

• For $C_{11}$: $$ C_{11} = 1 \cdot 1 + 2 \cdot 1 = 1 + 2 = 3 $$

• For $C_{12}$: $$ C_{12} = 1 \cdot 2 + 2 \cdot 1 = 2 + 2 = 4 $$

• For $C_{21}$: $$ C_{21} = 2 \cdot 1 + 1 \cdot 1 = 2 + 1 = 3 $$

• For $C_{22}$: $$ C_{22} = 2 \cdot 2 + 1 \cdot 1 = 4 + 1 = 5 $$

4. Compile results into the resulting matrix

The resulting matrix $C$ is:

$$ C = \begin{pmatrix} C_{11} & C_{12} \ C_{21} & C_{22} \ \end{pmatrix} = \begin{pmatrix} 3 & 4 \ 3 & 5 \ \end{pmatrix} $$