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Given the matrices:

[tex]\[ A=\left[\begin{array}{cc} 1 & -1 \\ 0 & 3 \end{array}\right], \quad B=\left[\begin{array}{cc} 0 & 2 \\ 1 & -1 \end{array}\right] \][/tex]

1. What is the size of the product [tex]\( C = AB \)[/tex]?

2. Find the product of [tex]\( D = BA \)[/tex]:
[tex]\[ D = \left[\begin{array}{cc} d_{11} & d_{12} \\ d_{21} & d_{22} \end{array}\right] \][/tex]

3. Find the product of [tex]\( C = AB \)[/tex].


Sagot :

Let's break down the given matrices and determine the results step by step as required.

### 1. Calculate the size of the product [tex]\( C = AB \)[/tex]

Given:
[tex]\[ A = \begin{pmatrix} 1 & -1 \\ 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 2 \\ 1 & -1 \end{pmatrix} \][/tex]

First, let's determine the size of the resulting matrix [tex]\( C \)[/tex]:
- Matrix [tex]\( A \)[/tex] is [tex]\( 2 \times 2 \)[/tex]
- Matrix [tex]\( B \)[/tex] is [tex]\( 2 \times 2 \)[/tex]

When multiplying two matrices, the resulting matrix has the number of rows of the first matrix and the number of columns of the second matrix. Thus, the product [tex]\( C = AB \)[/tex] will also be of size [tex]\( 2 \times 2 \)[/tex].

Therefore, the size of the product [tex]\( C \)[/tex] is:
[tex]\[ \boxed{(2, 2)} \][/tex]

### 2. Calculate the product [tex]\( D = BA \)[/tex]

Let's find the elements of the product [tex]\( D = BA \)[/tex]:
[tex]\[ B = \begin{pmatrix} 0 & 2 \\ 1 & -1 \end{pmatrix}, \quad A = \begin{pmatrix} 1 & -1 \\ 0 & 3 \end{pmatrix} \][/tex]

Now, we will compute each element of [tex]\( D \)[/tex]:

[tex]\[ d_{11} = (0 \cdot 1) + (2 \cdot 0) = 0 \][/tex]

[tex]\[ d_{12} = (0 \cdot -1) + (2 \cdot 3) = 6 \][/tex]

[tex]\[ d_{21} = (1 \cdot 1) + (-1 \cdot 0) = 1 \][/tex]

[tex]\[ d_{22} = (1 \cdot -1) + (-1 \cdot 3) = -4 \][/tex]

Therefore, the elements of [tex]\( D \)[/tex] are:
[tex]\[ \begin{aligned} d_{11} &= \boxed{0} \\ d_{12} &= \boxed{6} \\ d_{21} &= \boxed{1} \\ d_{22} &= \boxed{-4} \end{aligned} \][/tex]

### 3. Confirming the completion

Finally, the detailed elements of the required solution are:
- The size of the product [tex]\( C = AB \)[/tex] is [tex]\( \boxed{(2, 2)} \)[/tex].
- The elements of the product [tex]\( D \)[/tex] are [tex]\( d_{11} = \boxed{0} \)[/tex], [tex]\( d_{12} = \boxed{6} \)[/tex], [tex]\( d_{21} = \boxed{1} \)[/tex], and [tex]\( d_{22} = \boxed{-4} \)[/tex].

Hence, the values have been correctly identified according to the given tasks.
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