known: A=[-110120], B=213-102, C=[-1002], D=[02-12] Calculate the ABC of the matrix!

To calculate the product \( ABC \) of the matrices given, we first need to express each matrix properly. Based on your notation, let's define the matrices as follows:

1. \( A = \begin{bmatrix} -110 & 120 \end{bmatrix} \)
2. \( B = \begin{bmatrix} 213 & -102 \end{bmatrix} \)
3. \( C = \begin{bmatrix} -100 & 2 \end{bmatrix} \)
4. \( D = \begin{bmatrix} 0 & 2 \\ -1 & 2 \end{bmatrix} \)

However, since the operation requested is \( ABC \), we will only focus on multiplying \( A \), \( B \), and \( C \).

### Step 1: Multiply \( A \) and \( B \)

The multiplication of two matrices is defined as:

\[
(A \cdot B)_{ij} = \sum_k A_{ik} B_{kj}
\]

Since both \( A \) and \( B \) are row vectors (1x2), we can interpret \( B \) as a column vector to perform the multiplication.

So let's define:

\[
B = \begin{bmatrix} 213 \\ -102 \end{bmatrix}
\]

Now we multiply \( A \) with \( B \):

\[
A \cdot B = \begin{bmatrix} -110 & 120 \end{bmatrix} \begin{bmatrix} 213 \\ -102 \end{bmatrix} = (-110 \cdot 213) + (120 \cdot -102)
\]

Calculating that:

\[
= -23430 - 12240 = -35670
\]

### Step 2: Multiply the result with \( C \)

Next, we multiply the result \( -35670 \) with \( C \):

Since \( C \) is also a row vector:

\[
C = \begin{bmatrix} -100 & 2 \end{bmatrix}
\]

We can write the result as a scalar multiplication, since we are dealing with scalars.

\[
-35670 \cdot C = -35670 \cdot \begin{bmatrix} -100 & 2 \end{bmatrix} = \begin{bmatrix} 3567000 & -71340 \end{bmatrix}
\]

### Final Result

The final result of \( ABC \) is:

\[
ABC = \begin{bmatrix} 3567000 & -71340 \end{bmatrix}
\]

If you have any questions or need further clarification, feel free to ask! Here are some related questions you might consider:

1. What are the properties of matrix multiplication?
2. How can we represent these matrices in a different form (e.g., using Python)?
3. Can you explain how to multiply larger matrices?
4. What applications do matrix products have in real-world scenarios?
5. How does matrix multiplication differ from scalar multiplication? 

**Tip:** Always check the dimensions of matrices before attempting multiplication to ensure they are compatible!

Explore the detailed solution for calculating the product of matrices A, B, and C, including essential matrix multiplication formulas and compatibility rules. This problem highlights core concepts in matrix operations, suitable for students in Grades 10-12.

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