https://file.aiquickdraw.com/mathbot/file/17281883807004vtt4eqi.jpg

The problem in the image asks to determine the product \((BC)A\), where matrices \(A\), \(B\), and \(C\) are given as:

\[
A = \begin{bmatrix} -2 & 3 \\ -5 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} -5 & -8 \end{bmatrix}, \quad C = \begin{bmatrix} -1 & -3 \\ -4 & 10 \end{bmatrix}
\]

### Steps to solve:

1. **Dimensions Check**:
   - \(A\) is a \(2 \times 2\) matrix.
   - \(B\) is a \(1 \times 2\) matrix.
   - \(C\) is a \(2 \times 2\) matrix.

To compute \((BC)\), we need to check if the product \(BC\) is possible:
- \(B\) is \(1 \times 2\), and \(C\) is \(2 \times 2\), so their product is possible. The resulting matrix will be \(1 \times 2\).

2. **Compute \(BC\)**:
   \[
   BC = \begin{bmatrix} -5 & -8 \end{bmatrix} \begin{bmatrix} -1 & -3 \\ -4 & 10 \end{bmatrix}
   \]
   Perform the multiplication:
   \[
   BC = \begin{bmatrix} (-5)(-1) + (-8)(-4) & (-5)(-3) + (-8)(10) \end{bmatrix}
   \]
   \[
   BC = \begin{bmatrix} 5 + 32 & 15 - 80 \end{bmatrix}
   \]
   \[
   BC = \begin{bmatrix} 37 & -65 \end{bmatrix}
   \]

3. **Now, compute \((BC)A\)**:
   - \(BC\) is a \(1 \times 2\) matrix, and \(A\) is a \(2 \times 2\) matrix. The product \((BC)A\) is possible, and the resulting matrix will be \(1 \times 2\).

   Now, multiply \(BC\) with \(A\):
   \[
   (BC)A = \begin{bmatrix} 37 & -65 \end{bmatrix} \begin{bmatrix} -2 & 3 \\ -5 & 3 \end{bmatrix}
   \]
   Perform the multiplication:
   \[
   (BC)A = \begin{bmatrix} (37)(-2) + (-65)(-5) & (37)(3) + (-65)(3) \end{bmatrix}
   \]
   \[
   (BC)A = \begin{bmatrix} -74 + 325 & 111 - 195 \end{bmatrix}
   \]
   \[
   (BC)A = \begin{bmatrix} 251 & -84 \end{bmatrix}
   \]

Thus, \((BC)A = \begin{bmatrix} 251 & -84 \end{bmatrix}\).

### Result:
The product \((BC)A\) is possible, and the resulting matrix is \(\begin{bmatrix} 251 & -84 \end{bmatrix}\).

---

Would you like more details or have any questions?

Here are five related questions:
1. What are the conditions for matrix multiplication to be possible?
2. How do you find the dimensions of the resulting matrix after multiplication?
3. What properties of matrices help in simplifying matrix products?
4. Can you describe the steps to multiply two matrices by hand?
5. What happens if you change the order of multiplication in matrices?

**Tip:** Remember, matrix multiplication is not commutative! \( AB \neq BA \) in general.

Given the matrices A = [[-2, 3], [-5, 3]], B = [[-5, -8]], and C = [[-1, -3], [-4, 10]], determine if (BC)A is possible and calculate the resulting matrix.

Learn how to determine if the matrix multiplication (BC)A is possible for matrices A = [[-2, 3], [-5, 3]], B = [[-5, -8]], and C = [[-1, -3], [-4, 10]]. This detailed solution includes matrix dimensionality checks, step-by-step calculations, and the resulting matrix.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation